THREE ESSAYS ON THE DYNAMIC ANALYSIS OF DEMAND FOR FACTORS OF PRODUCTION by CATHERINE JOAN MORRISON B.A., The Un ivers i t y of B r i t i s h Columbia, 1977 M.A., The Un ivers i t y of B r i t i s h Columbia, 1978 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 t h i s thes i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1982 © C a t h e r i n e Joan Morr ison, 1982 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 DE-6 (3/81) i Three Essays in the Dynamic Ana lys i s of Demand for Factors of Product ion ABSTRACT Th is d i s se r t a t i on cons i s t s of three essays focuss ing on var ious theo re t i c a l extensions and empir ica l implementations of a model cha r a c t e r i z i ng the dynamic input demand behavior of the f i rm . The ana l ys i s i s based on recent developments on optimal investment dec is ions over time subject to i nc reas ing marginal costs of adjustment. Theoret ica l extensions to the model inc lude the incorporat ion of ( i ) monopoly behavior based on p r o f i t maximization with both in te rna l and external costs of adjustment on gross investment (Essay 1) and ( i i ) non-stat ic output and p r i ce expectat ions (Essay 2). Both theore t i ca l models are emp i r i c a l l y implemented using annual U.S. manufacturing data, 1948-77. Results from the var ious s p e c i f i c a t i o n s can there fore be analyzed and con t r as t ed . Essay 3 cons i s t s of an app l i c a t i on of the a l t e rna t i ve models to measures of capac i ty u t i l i z a t i o n s p e c i f i e d i n terms of the e x p l i c i t dynamic opt imizat ion model of the f i r m ' s behavior . Various de r i va t i ons and in te rp re ta t ions o f these measures fo r a l t e rna t i v e theo re t i c a l s p e c i f i c a t i o n s are developed and compared, both a n a l y t i c a l l y and e m p i r i c a l l y . The three essays are l inked by a common s t ruc tura l model which provides a bas is fo r incorporat ion o f d i f f e r e n t behavioral assumptions, and fo r examination and comparison of the various models in terms of demand e l a s t i c i t i e s , c y c l i c a l i nd i ca to r s such as mu l t i f a c to r p roduc t i v i t y and Tob in ' s " q " , and capac i ty u t i l i z a t i o n . Cather ine J . Morrison i i Table of Contents Page Introduct ion 1 Essay 1. Models o f Investment with Endogenous Output and Imperfect Competition I. Introduct ion 9 II. Ex i s t i ng L i t e r a tu r e : A Review of S t a t i c and Dynamic Modeling o f Investment Dec is ions 11 III. The Monopoly Model 25 IV. Empir ica l Spec i f i c a t i on and Implementation IV.A. Data 43 IV. B. Econometric Considerat ions 43 V. Empir ical Results V. A. Est imat ion and Parameter Estimates 52 V.B. Costs of Adjustment 59 V .C . Monopoly 75 VI. Concluding Remarks 87 Footnotes 89 i i i Essay 2. A S t ruc tura l Model o f Dynamic Factor Demands with Non-Static Expectat ions I. Introduct ion 93 II. A Review of the L i te ra ture II.A. Dynamics 96 II.B. The F i rm's Investment Decis ions with Non-Static Expectat ions 100 II. C. The Non-Static Expectat ions Framework 102 III. The Model: Theoret ica l Foundations 107 III. A. The Determin is t ic Opt imizat ion Problem with Non-Static Expectat ions 107 III.B. A l te rna t i ve Ana l y t i ca l Spec i f i c a t i ons for Expectat ions Formation 115 11I.C. Empir ical Spec i f i c a t i on 123 IV. Econometric Implementation 127 V. Empir ical Results 149 VI. Concluding Remarks 165 Footnotes 168 Essay 3. On the Economic In terpreta t ion and Measurement of Optimal Capacity U t i l i z a t i o n I. Introduct ion II. Economic Measures o f Capacity U t i l i z a t i o n and Other Cyc l i c a l Ind ica tors : An Overview 172 177 i v III. Graphical Der ivat ion and Interpretat ion of Economic Capacity U t i l i z a t i o n Measures: Extensions to the Monopoly and Non-Static Expectations Models 185 I I I .A. P r o f i t Maximization and CU Measurement 187 III.B. Non-Static Expectat ions and CU Measurement 193 IV. Issues i n CU Measurement: A More Formal Ana l y t i c a l Treatment 208 V. Empir ica l Results 229 VI. Concluding Remarks 246 Footnotes 251 Concluding Remarks and Suggestions fo r Further Research 256 Bibl iography 261 Appendices Appendix A: Forecast ing with Economic Time Ser ies and the "Rat ional Expectat ions" Hypothesis 268 Appendix B: The Convergence Approach 276 Appendix C: Data 280 Appendix D: ARIMA Results for the Adaptive Expectations Model of Essay 2 287 V L i s t of Tables Page 1-1. Parameter Estimates with A l t e rna t i ve Spec i f i c a t i ons on Investment and Adjustment Costs , 1948-77. 53 1-2. Parameter Estimates for Selected Spec i f i c a t i ons on Investment and Adjustment Costs , 1948-71. 60 1-3. Output Pr ice and Demand E l a s t i c i t i e s for A l t e rna t i ve Spec i f i c a t i ons on Investment and Adjustment Costs , 1948-77. 77 1-4. Pr ice E l a s t i c i t y Estimates for A l t e rna t i ve Investment and Adjustment Cost S p e c i f i c a t i o n s , 1948-77. 85 11-1 - Parameter Est imates, A l t e rna t i v e Expectations S p e c i f i c a t i o n s , 1949-76. 151 11-2. Parameter Est imates, A l t e rna t i ve Expectations S p e c i f i c a t i o n s , 1949-71. 154 11-3. P r i ce and Output E l a s t i c i t y Est imates, A l t e rna t i ve Expectat ions S p e c i f i c a t i o n s , 1949-76. 159 I I I-l. C y c l i c a l Measures, T rad i t i ona l and Economic 175 111-2. Capacity U t i l i z a t i o n Quantity and Cost-Based Indices, A l t e rna t i v e Investment and Adjustment Cost S p e c i f i c a t i o n s , " Fu l l Adjustment" Monopoly Model, 1954-77 232 111-3. Capacity U t i l i z a t i o n Quantity and Cost-Based Indices , A l t e rna t i v e Investment and Adjustment Cost S p e c i f i c a t i o n s , " S t a t i c Short Run" Monopoly Model, 1954-77 235 111-4. Capacity U t i l i z a t i o n Quantity and Cost-Based Indices, A l t e rna t i ve Expectat ions S p e c i f i c a t i o n s , General Dynamic Model, 1954-76. 238 111-5 Capacity U t i l i z a t i o n Quantity and Cost-Based Indices, A l t e rna t i ve Expectat ions S p e c i f i c a t i o n s , S t a t i c Approach, 1954-76. 239 L i s t o f F igures Page F igure I-l Internal and External Costs of Adjustment on Gross Investment 68 Figure 1-2 Total Costs of Adjustment on Gross Investment 69 Figure 1-3 Internal Costs of Adjustment, Internal Costs Only Model 70 Figure 1-4 External Costs of Adjustment, External Costs Only Model 71 Figure 1-5 Internal Costs o f Adjustment, Net Investment Model 72 Figure III-l CU Representat ion, Cost Minimizat ion Case 186 Figure II1-2 CU, P r o f i t Maximization Case 186 Figure III-3 CU, Monopoly Case (a) 190 Figure 111-4 CU, Monopoly Case (b) 190 Figure 111-5 Investment Paths Resul t ing from Various Expectat ions about an Output Demand Increase 196 Figure 111-6 CU, Non-Static Expectat ions Cost Minimizat ion Case 200 Figure III-7 Paths of Output and Capi ta l Resul t ing from a Dynamic Opt imizat ion Model for S ta t i c and Non-Static Expectations Assumptions 200 Figure III-8 Qua l i t a t i v e Patterns of Cyc l i c a l Measures from Dynamic Opt imizat ion of a Firm Facing a Demand Increase 204 Figure 111-9 Shadow and Total Cost Functions (a) 216 Figure III-10 Shadow and Total Cost Functions (b) 216 v i i Acknowledgements The completion of a long-term pro jec t l i k e t h i s thes is and the degree i t represents i s a f fec ted by many people in many ways. This one i s no except ion. I f i r s t want to thank Keizo Nagatani, my adv isor , for encouragement and help in completing th i s d i s s e r t a t i o n . I would a lso l i k e to thank the other members of my committee, John He l l iwe l l and espec i a l l y Erwin Diewert f o r support at important times along the way. Other people who have contr ibuted to the development of t h i s d i s se r t a t i on are David 0. Wood at the M.I.T. Energy Laboratory, who allowed me much freedom to overrun the Energy Lab, J . Randolph Norsworthy, who has been a good working partner and f r i end s ince I worked with him at B.L.S. in Washington D .C . , M. Ishaq N a d i r i , who contr ibuted he lpfu l comments and suggestions while my work was proceeding during my year at New York Un i ve r s i t y , and Stephen Nickel 1 at L . S . E . , who answered my wr i t ten plea for help at a stopping point in my mathematical ana lys i s with ass i s tance that allowed me to keep going. I want to thank a l l of these people fo r t he i r help and support. There are a lso many important people who r e a l l y contr ibuted of themselves, who re jo i ced with me when the way I chose to go seemed to be b r igh t and f u l f i l l i n g , and helped me keep going when the path turned out , at t imes, to be d i f f i c u l t and d iscourag ing. I w i l l always be indebted to them for t he i r support , encouragement, and he lp ; I want to thank them e s p e c i a l l y . The f i r s t people to thank—who persevered with me through everything—are the members o f my fami ly . My parents , Roy and P h y l l i s Morr ison, and my s i s t e r s , Debbie and Barbara have been my a l l i e s a l l along the way. The second v i i i i s my previous husband, Gair White. He has cheered my progress since I f i r s t s ta r ted Un ivers i t y and provided me with support , encouragement, and b e l i e f in me and in l i f e that has been a foundation for everything I have been able to accompl ish. Over and above the foundation of support and love that these members of my family have provided me, however, Ernie Berndt, my f r i e n d , companion, and working par tner , has helped me develop knowledge of myself and my goals along with b e l i e f in my c a p a b i l i t i e s that has allowed me to persevere and look forward to our fu tu re . Specia l thanks go to him. Without these people the years spent working on th i s degree and th i s thes i s would have been fa r more d i f f i c u l t . I am thankful for them a l l . -1-Introduct ion Since A l f r e d Marshall introduced the theore t i ca l d i s t i n c t i o n between the short and long-run behavior o f the f i r m , i t has been understood that own pr i ce responses are l i k e l y to be smal ler in the short than in the long run. More r ecen t l y , s tudies in the framework o f comparative dynamics have noted that the "impact" e f f e c t s of f l uc tua t ions or shocks on dec is ion var iab les of the f i rm are often very d i f f e r e n t from long run e f f e c t s . It i s c l e a r that dynamic models e x p l i c i t l y spec i f y ing the short and long run and the f i r m ' s adjustment process as r e su l t s of a f i rm ' s integrated dec is ion process are c ru c i a l to an understanding of fac tors a f f e c t i ng demand for inputs . In the past decade, researchers have attempted to incorporate the Marsha l l ian framework in to both theore t i ca l and empir ical analyses. Theoret ica l s tudies have st ressed the incorporat ion of f ixed stocks and endogenous adjustment processes r e su l t i ng from opt imizat ion by the f irm given these stock (or other f ixed) v a r i ab l e s . * Attempts to impose th i s s t ruc ture on econometric models, however, have genera l ly taken the form of ad hoc lag s t ruc tu res , often in the form of a " p a r t i a l adjustment" s p e c i f i c a t i o n imposed on e s s e n t i a l l y s t a t i c models. Because o f the stock nature and the r e su l t i ng " f i x i t y " of c a p i t a l , the ana lys i s o f investment dec is ions provides an important example o f how theor i s t s and econometricians have formulated problems in modeling dynamic input demands. These studies have often been based on models formulated to determine an optimal or "des i r ed " level of cap i t a l s tock. Current ly there are 2 two dominant investment theo r i e s , the " q " theory o f investment , and the " neo-c l a s s i c a l " or " Jorgensonian" theory. Investment theor ies based on these -2-models have evolved by imputing various assumptions concerning adjustment processes to these "des i red " l e v e l s . The " q " theory o f investment can be summarized as fo l lows : The f i rm maximizes i t s present worth as given by the market value of i t s s e c u r i t i e s , and investment i s undertaken i f i t increases the value of these s e c u r i t i e s . I f the value of the pro jec t as determined by the s e cu r i t i e s market exceeds the cos t , investment w i l l be undertaken and increases in the share p r i ce w i l l occur . Thus, "the rate of investment—the speed at which investors want to increase the cap i t a l s tock—should be r e l a t e d , i f to anything, to q, the value of c ap i t a l r e l a t i v e to i t s replacement c o s t " , such that values of q above one 3 encourage investment (and converse ly ) . The " Jorgensonian" approach i s based on a d i f f e r e n t type of ana l y s i s , but the i n t e rp re ta t i on i s analogous. In Jorgenson's theory, "des i r ed " c ap i t a l stock i s determined by a model of the f i rm ' s opt imizat ion process which equates marginal benef i t s and costs of increments in the cap i t a l s tock, and investment then i s determined as a proport ion of the discrepancy between the des i red and current cap i t a l s tock. The adjustment proport ion presumably can be determined by adjustment costs assoc iated with investment, but these costs are not e x p l i c i t l y taken in to account in the f i r m ' s investment dec i s i ons , so determination of the rate of investment i s e s s e n t i a l l y ad hoc. The q and neoc lass i ca l investment theor ies depend, there fore , on two concepts which have s im i l a r i n t e rp re t a t i ons . In the one the notion of a "des i r ed " c ap i t a l stock i s determined by the comparison of the shadow pr i ce and the market value of c a p i t a l ; in the other i t i s determined by the comparison of the marginal costs and benef i t s of c a p i t a l . However, determination of the rate of investment depends on an e x p l i c i t model of the f i rm ' s response to t h i s d i s equ i l i b r i um . Michael Love l l therefore c r i t i c i z e s -3-both of these theor ies on the same grounds: "Nei ther approach works out the dynamics of the adjustment process with in the context of a c a r e f u l l y a r t i c u l a t e d opt imizat ion framework that would s p e c i f i c a l l y incorporate the process of expectat ion formation and adjustment c o s t s . In t h i s respect , both theor ies are dominated by a number o f con t r ibu t ions that have der ived the optimal time path of adjustment simultaneously with the determination of the proper t ies of the ult imate long-run equ i l i b r i um . " (as quoted in Yoshikawa (1980)). Both " q " theor ies and pa r t i a l adjustment neoc lass i ca l theor ies can be der ived from the models of the f i r m ' s integrated dec is ion process formulated by Treadway, Lucas and Mortenson. These models e x p l i c i t l y take account of inc reas ing costs of adjustment assoc iated with investment and therefore determine the rate of investment in quas i-f ixed fac tors in an integrated opt imizat ion framework with an endogenous adjustment process from the short to the long run. These theore t i ca l models have been the basis of recent econometric s tudies which al low for endogenous stock adjustment and 4 i n t e r r e l a t ed demands fo r va r iab le f ac to r demands. Although these dynamic models e x p l i c i t l y descr ibe the short run-long run d i s t i n c t i o n , the adjustment process in response to permanent changes, and are therefore useful for ana lys i s of f l uc tua t ions and c y c l i c a l behavior of the economy, in many ways at present they remain too r e s t r i c t e d for ana lys i s o f temporary changes and shocks. For example, these models do not al low fo r the f i r m ' s f l e x i b i l i t y in response to shocks or cyc les in terms of output inventory or p r i ce changes. They a lso do not take in to cons iderat ion var ious important determinants o f the stock behavior o f the f i r m , e . g . , p r i ce and output expectat ions formation. Some studies have attempted to incorporate these types of f l e x i b i l i t y in to t r ad i t i ona l de te rmin is t i c models by imposing 5 complicated s tochast i c processes onto r e l a t i v e l y simple s t ruc tura l models. However, these c h a r a c t e r i s t i c s o f the dec is ion making process could in -4-p r i n c i p l e a lso be represented in terms of de te rmin is t i c s t ruc tura l processes; the f i r m ' s behavioral formulat ion should preferab ly e x p l i c i t l y permit buf fers such as output pr i ce changes or u t i l i z a t i o n rates as dec is ion v a r i ab l e s , and non-stat ic expectat ions of the future as dec is ion determinants. In t h i s d i s se r t a t i on I emphasize the buf fers the f i rm uses in response to f l u c tua t i ons , the impl i ca t ions of e x p l i c i t incorporat ion of these stocks fo r such dec is ion processes as expectat ions formation and pr i ce s e t t i n g , and the r e su l t i ng impact on cha rac te r i za t i on of economic i nd i ca to r s of c y c l i c a l behavior such as capac i ty u t i l i z a t i o n . I therefore extend both the theore t i ca l and empir ica l l i t e r a t u r e on the f i r m ' s production behavior to permit e x p l i c i t cons iderat ion of the i n t e r r e l a t i onsh ips among these dec is ion var iab les and t h e i r e f f e c t s . To develop t h i s extended model, I u t i l i z e as the "base" or " s ta te of the a r t " framework an integrated dynamic opt imizat ion model of the f i rm spec i f i ed t h e o r e t i c a l l y by Treadway, Lucas, and others , and developed for empir ica l implementation by Fuss (1976) and Berndt, Fuss and Waverman (1977, 1979). Genera l iza t ions are incorporated to al low fo r add i t iona l f l e x i b i l i t y in response to exogenous shocks. These refinements are based on more complete cons iderat ion of input and output stocks and the r e su l t i ng importance of other dec i s ion processes such as expectat ions format ion, output p r i ce determinat ion, and labor and cap i t a l u t i l i z a t i o n . Th is a l so requires e x p l i c i t l y taking account of the convexity of the marginal cos t funct ion fo r adjustment of quas i- f ixed f a c t o r s , which i s required fo r slow adjustment of s tocks . Due to the complexity of the theore t i ca l model, only a l im i t ed number o f refinements can be incorporated in to the model at one time while s t i l l r e t a in ing theore t i ca l and empir ica l t r a c t a b i l i t y . Thus I develop two extensions of the "base model" of the f i rm ' s dec is ion process , which I be l ieve -5-capture important empir ica l and theore t i ca l extensions to the f i r m ' s integrated behavioral process. A t h i r d extension emphasizes the use of dynamic models to provide micro-foundations fo r macro-economic concepts. Th is thes i s i s therefore s t ructured as three se l f-conta ined essays, each based on the theore t i ca l and empir ica l development of one of these extens ions. Econometric r e su l t s fo r each essay der ive from est imat ion o f the model using annual U.S. manufacturing data for 1947-77. Due to the aggregated nature of the data these r esu l t s are meant to be i l l u s t r a t i v e rather than conc lus i ve . The f i r s t essay emphasizes the progression from s t a t i c to inc reas ing ly complex dynamic models, and then focuses on two main issues l a rge l y neglected in the "s ta te-of-the-ar t " l i t e r a t u r e : ( i ) the existence of convex costs of adjustment which are necessary fo r using an adjustment costs approach as a foundation fo r investment ana l y s i s , and ( i i ) recogni t ion of imperfect ly compet i t ive aspects of market s t ructure in investment models. I f i nd that in te rna l costs o f adjustment tend to r e f l e c t the convex shape necessary fo r adjustment costs to impose a cons t ra in t on rap id accumulation of quas i- f ixed inputs , whereas external costs appear s l i g h t l y concave. I am not, however, able to r e j e c t the hypothesis of zero external costs of adjustment, implying a neg l i g i b l e monopsonistic impact in the cap i t a l market. I a lso f i nd that imperfect ly compet i t ive behavior in the output market, as charac te r ized by a monopoly model, appears to provide s i g n i f i c a n t add i t iona l explanatory power for f i rm behavior. In the second essay I const ruct a s t ruc tura l model incorporat ing non-stat ic expectat ions o f future output demand and r e l a t i v e input p r i c e s , so that the f i rm faces a " ta rge t " stock leve l o f i t s cap i t a l stock which i s the "des i r ed " cap i t a l stock given current values of the exogenous var iab les plus a -6-weighted average o f future changes in exogenous v a r i ab l e s . This framework impl ies a s p e c i f i c s t ructure which can be used as the bas is of an emp i r i c a l l y implementable model. The general form can be used to represent a form of " pa r t i a l r a t i o n a l i t y " , where the parameters of the expectat ions process are not e x p l i c i t l y i d e n t i f i e d but a l l future expectat ions are captured in composite parameters. A l t e r n a t i v e l y , t h i s general form can be r e s t r i c t e d to adaptive expectat ions . In the l a t t e r case the investment equation has an e x p l i c i t s t ructure depending on the parameters of both the technology and the expectat ions formation process. I f i nd that the adaptive expectat ions model appears to r e f l e c t the important components o f non-stat ic expectat ions of the time paths of future exogenous va r i ab l e s , while imposing s u f f i c i e n t s t ructure on the data to derive reasonable behavior impl i ca t ions for the f i rm . It i s a lso quite successful emp i r i c a l l y . In the t h i r d essay I apply these dynamic models to the measurement of c y c l i c a l i nd i ca to r s such as p roduc t i v i t y and Tob in ' s q , and p a r t i c u l a r l y capac i ty u t i l i z a t i o n . I emphasize the generation of these ind i ca to rs as e x p l i c i t r e su l t s of the f i r m ' s dynamic opt imizat ion process , and the impl i ca t ions for using these micro foundations to const ruct t h e o r e t i c a l l y based capac i ty u t i l i z a t o n ind ices dependent on the existence of costs of adjustment. I f i nd that ind ices based on the dev iat ion of stock var iab les from t h e i r long run equ i l ib r ium values can be der i ved , and ind ica te how theore t i ca l incorporat ion of var ious components of the dynamic models, e . g . , non-stat ic expectat ions , w i l l a f f e c t the measures. I then ca l cu l a t e capac i ty u t i l i z a t i o n ind ices based both on the optimal as compared to the cur rent l y demanded output leve l--a quantity i nd i ca to r corresponding to the usual concept o f capac i ty u t i l i z a t i o n measurement. I a l so der ive and estimate a dual cost i nd i ca to r which r e f l e c t s the costs o f being away from the "opt ima l " output l e v e l . -7-The f i na l chapter of t h i s d i s se r t a t i on contains concluding remarks, suggestions fo r fur ther research, and appendices. Emphasis throughout the d i s se r t a t i on i s on the in tegra t ion of var ious bodies of l i t e r a t u r e , on f ac to r demand, investment, Tob in ' s q, and capac i ty u t i l i z a t i o n in both theore t i ca l and empir ica l contexts , in to a framework in which the i n t e r r e l a t i onsh ip s among these economic phenomena can be der i ved , c l a r i f i e d and in te rp re ted . -8-FOOTNOTES, Introduct ion 1 See for example Treadway (1971, 1974), Lucas (1967, 1969), and Mortensen (1973). 2 Tobin (1961, 1969). 3 See Yoshikawa (1980), quote from Tobin (1969). 4 See, e . g . , Fuss (1976), Berndt, Fuss and Waverman (1977, 1979, 1980), Morrison and Berndt (1981), and Epstein and Denny (1980). 5 See, for example, Eichenbaum (1980), and Green and Laf font (1980). -9-Essay 1. Models of Investment with Endogenous Output and Imperfect Competition I. Introduct ion In the past decade numerous cont r ibut ions to the theory of the behavior of the f i rm have appeared in the l i t e r a t u r e , a number of which have incorporated dynamic aspects in to what t r a d i t i o n a l l y have been s t a t i c models.* Recognit ion of intertemporal e f f e c t s has been p a r t i c u l a r l y s i g n i f i c a n t fo r the theory of investment demand. The word " investment" i t s e l f impl ies some l i nk between the present and the future in the sense that " . . . the act of investment involves the a cqu i s i t i on of a good which i s dest ined not to be consumed or e n t i r e l y used up in the current p e r i o d " . Most o f the recent theore t i ca l work on investment has been d i rec ted toward gaining new ins igh ts in to the impl i ca t ions of t h i s time-dependency on decision-making. In genera l , ana lys i s has focused on exp la in ing why the f i rm ' s adjustment of i t s cap i t a l stock i s slow rather than instantaneous, contrary to pred ic t ions based on s t a t i c models. I be l ieve i t i s important to examine these recent developments c r i t i c a l l y , c l a r i f y i n g how a p r i o r i r e s t r i c t i o n s imbedded in e a r l i e r models have been genera l ized to more r e a l i s t i c assumptions, and then de l inea t ing under ly ing assumptions that s t i l l remain. Recent surveys of t h i s type inc lude Brech l ing (1975) and N i cke l l (1978). Th i s Essay i s d i rec ted along these l i n e s . Its purpose i s to ind ica te how a basic dynamic model of investment dec is ions can use fu l l y be genera l i zed , and to assess the importance of remaining r e s t r i c t i o n s . To pursue t h i s goa l , i n the next sect ion I present a b r i e f survey designed to h igh l igh t aspects of -10-a l t e rna t i v e emp i r i ca l l y t rac tab le models incorporat ing fewer r e s t r i c t i v e hypotheses. I f i r s t consider a very simple model of the f i rm. I then b r i e f l y d iscuss recent genera l i za t ions in terms of a " s ta te of the a r t " bas ic dynamic model of Fuss (1976), and Berndt, Fuss and Waverman (BFW) (1979) which incorporates dynamic opt imizat ion behavior e x p l i c t l y into the model. In the t h i r d sect ion I present a new theore t i ca l extension of BFW (1979) and Morrison-Berndt (1981)—a monopoly model—which re ta ins the a t t r a c t i v e feature of being emp i r i ca l l y implementable. Although a number of r e s t r i c t i o n s are re laxed in t h i s model, some important cons t ra in ts remain, p a r t i c u l a r l y those invo l v ing non-stat ic expectat ions and uncer ta inty . In the fourth sect ion I d iscuss problems with econometric s p e c i f i c a t i o n , while in the f i f t h sect ion I present empir ica l r e su l t s on the monopoly model developed in sect ion III, using annual data for the U.S. manufacturing industry 1947-77. More r e s t r i c t i v e cases of the model are a lso d iscussed , some of which involve tes tab le parametric r e s t r i c t i o n s . Some major conclus ions are that costs of adjustment appear bet ter def ined over gross than net investment, and that in terna l costs incorporate the convexity necessary for slow investment behavior whereas external costs do not. However, to ta l costs of adjustment are convex. In add i t i on , according to var ious i n d i c a t o r s , the monopoly s p e c i f i c a t i o n appears to r e f l e c t more p rec i se l y important c h a r a c t e r i s t i c s of the behavior of f irms in the manufacturing sector than does a competi t ive model. A f i n a l sect ion sums up the var ious propos i t ions discussed in the paper. -11-II. The Ex i s t i ng L i t e r a tu r e : A Review of S ta t i c and Dynamic Modeling o f Investment Decis ions T rad i t i ona l s t a t i c models of investment dec is ions incorporate numerous s t r ingent s imp l i f y i ng assumptions. S ix t yp i ca l fundamental assumptions have Assumption 1: There ex i s t s a per fec t cap i t a l market. Assumption 2: The world i s one of per fec t ce r t a in t y concerning the future (or equ iva lent l y a l l i nd i v idua l s hold the same ce r t a in expectat ions about the fu tu re ) . Assumption 3: The f i rm employs only two fac tors o f product ion , c ap i t a l and labor , which i t uses to produce a s ing le output. E f f i c i e n t production techniques can be summarized in the form of a twice d i f f e r e n t i a t e production funct ion with pos i t i v e marginal products and s t r i c t l y decreasing returns to scale everywhere. Assumption 4: Capi ta l has the same productive c h a r a c t e r i s t i c s whatever i t s age or b i r thda te ; however, as i t ages, i t de ter io ra tes at a constant exponential ra te . Assumption 5: The f i rm acts as a pr i ce taker in a l l markets. In p a r t i c u l a r , cap i t a l can be bought or so ld in any quanti ty and becomes productive immediately upon purchase. Assumption 6: There are no costs involved e i the r in the sale and purchase of cap i t a l goods or in the productive implementation of new c a p i t a l . S ince the f i rm invests in cap i t a l stock to provide i t s e l f with c ap i t a l serv ices fo r the production process, the f i rm ' s investment dec is ion i s analyzed with in the framework of an input demand model. The usual formulat ion of t h i s type of model i s that the f i rmjnaximizes the present value of the been : income stream produced over t ime, V= I e R ( t ) , where R(t) i s : 1.2.1) R(t) = p(t) F (K ( t ) , L ( t ) ) - w(t)L(t ) - a ( t ) Z ( t ) , subject t o : -12-1.2.2) F (K ( t ) , L ( t ) )=Y( t ) , K(t) = Z( t )-dK(t ) , K(0) = K Q , where K(t) i s the cap i t a l stock employed at time t , L ( t ) i s the labor input , Z(t) i s the gross investment in new cap i ta l stock, p(t) i s the pr i ce of output, a(t) i s the asset pr i ce of cap i t a l goods, w(t) i s the wage ra te , r ( t ) i s the rate of i n t e r e s t , d i s the exponential rate of deprec ia t ion of the cap i t a l stock, Y(t) i s output, and F( ) i s the production func t ion . The opt imiz ing production plan corresponds to a sequence of K(t) and L( t ) demands sa t i s f y i ng the cond i t i ons : 1.2.3) p(t) 3F/3l_(K(t),L(t)) = w(t) fo r a l l t>0, 1.2.4) p(t) 3F/aK(K(t ) ,L ( t ) ) = a ( t ) ( r ( t )+d-a( t )/a ( t ) ) , f o r a l l t>0, which represent the usual optimal cond i t i ons , MRPL=w and MRPK= the rental cos t o f c a p i t a l = u ( t ) T h e s e equations can be solved to y i e l d demand funct ions K*(t) and L*(t ) , and then, from the cons t ra in t l i nk i ng Z to K, Z( t ) defined in terms of K*(t) can be expressed as Z* ( t ) , a l l of which depend on current values of v a r i ab l e s . The problem i s e s sen t i a l l y s t a t i c , even though i t would appear to be dynamic, because of the s t r ingent s imp l i f y ing assumptions. In f a c t , the necessary condi t ions here are the same as the necessary condi t ions fo r the problem of maximizing instantaneous p r o f i t s . This i s a r e su l t of the assumptions of constant exponential decay, the homogeneity of c a p i t a l , and instantaneous and cos t l e ss adjustment of c a p i t a l , which together imply that past dec is ions of the f i rm do not const ra in present dec i s ions . Even the assumption of ce r ta in t y does not impose problems here, because cap i t a l can be f u l l y and instantaneously adjusted when the future -13-a r r i v e s . I t i s therefore c l e a r that these assumptions suggest very u n r e a l i s t i c behavioral imp l i ca t ions for the model. One of the f i r s t assumptions that might be relaxed i s that of cos t l e s s and instantaneous adjustment. Bas i ca l l y there are two d i f f e r e n t assumptions incorporated in t h i s r e s t r i c t i o n , implying a l t e rna t i ve ways to extend the ana l y s i s . One can cons ider incorpora t ing costs o f adjustment fo r "quas i - f i xed " f a c t o r s—fac to r s f ixed in the short run but var iab le in the long run. The f i r s t imp l i ca t ion of th i s i s that one must spec i fy which input(s) are quas i - f i xed . In a d d i t i o n , one may want to cons ider a l t e rna t i v e inputs that can be choice var iab les fo r the f i r m . Inputs other than K and L which may be important components of the f i r m ' s production process inc lude energy (E) and non-energy intermediate mater ia ls (M). As a representat ion of t h i s gene ra l i z a t i on , one can wr i te the short run production funct ion as Y=F (x ,v , t ) , where x i s a vector o f I quas i- f ixed inputs , and v i s a vector o f J va r i ab le inputs . Now assume, as has been popular in the recent l i t e r a t u r e , that there are e x p l i c i t adjustment costs assoc iated with net investment for each of the quas i- f ixed inputs . These adjustment costs increase at an increas ing rate with the absolute s i ze of the amount of investment, and are zero only when net investment i s zero . Such a s p e c i f i c a t i o n impl ies ce r t a in addi t iona l assumptions. Although pr i ces are f i xed (as assumed be fore ) , adjustment costs 5 are p o s i t i v e and convex, and depend on net investment. Nei ther o f these l a t t e r two assumptions i s a p r i o r i obvious, and therefore below I cons ider them in greater d e t a i l . I f adjustment costs are incorporated in to the present value maximization formula t ion , the necessary cond i t ion fo r c ap i t a l changes s l i g h t l y from the f i r s t case considered above. S p e c i f i c a l l y , although the optimal production plan involves i n s t a l l i n g an extra un i t of cap i t a l stock un t i l i t s marginal -14-cost j u s t equates the marginal bene f i t , ana l y t i ca l expressions o f costs and benef i t s involved become considerably more complex.** The net gain becomes a funct ion of the complete path of c ap i t a l through time, which in turn i s a funct ion of the cur rent rate o f investment. Since adjustment cos ts l i nk the future to present dec is ions on investment, t h i s formulation requi res an assumption concerning expectat ions of the fu ture . I f one incorporates convex costs o f adjustment and assumes constant ( r e l a t i ve ) p r i ces one can der ive the f l e x i b l e acce le ra tor model of net investment. The "des i r ed " quant i ty of c ap i t a l stock i s e a s i l y determined as the long run equ i l ib r ium leve l o f c a p i t a l , given p r i c e s . Since inc reas ing costs of adjustment e x i s t , only a port ion of the d i f f e rence between the des i red and actual c ap i t a l stock w i l l be reduced over a short time i n t e r v a l . The assumption of constant r e l a t i v e p r i c e s , which impl ies myopic ( s t a t i c ) expectat ions by the f i r m , i s c ruc i a l to t h i s f l e x i b l e acce le ra tor behavioral r e l a t i o n s h i p . A problem a r i ses when r e l a t i v e pr i ces are allowed to change, because in such a case K* i s not we l l-def ined ; a time path of K* i s generated with d i f f e r e n t values dependent on expectat ions concerning pr i ces at each po in t in time. The above d iscuss ion ind ica tes that the dynamic model with increas ing marginal costs o f adjustment generates r i che r r e su l t s than the bas ic s t a t i c model.^ However, s ince a l t e rna t i ve assumptions could poss ib l y generate s im i l a r r e s u l t s , one must look at the i m p l i c i t assumptions more c l o s e l y . The assumption of increas ing marginal external costs of adjustment appears to be a p r i o r i p l aus ib l e as postulated in Keynes' General Theory. I f one assumes external costs are a s i g n i f i c a n t por t ion of to ta l adjustment c o s t s , then the convexity assumption seems reasonable. S p e c i f i c a l l y , the concept of external adjustment costs i s der ived from the idea of a "premium" that must be paid to obtain new cap i t a l goods. I f the f i rm ' s demand fo r c ap i t a l i s large -15-compared to to ta l market cap i t a l demand (which would be the case i f c ap i t a l i s s p e c i f i c to f i r m s ) , then convex external costs of adjustment may be reasonable. Thus t h i s idea impl ies a monopsonistic element; the f i rm i s not a p r i ce taker in the c a p i t a l market. A l t e r n a t i v e l y , there may a lso be in terna l costs of adjustment, which again may or may not be convex. Brechl ing and Mortensen (1971) def ine interna l costs as " . . . equiva lent to the propos i t ion that the inputs used by the f irm at one po int in time are at l eas t p a r t i a l l y 'produced' by the f i rm at some e a r l i e r da te " . Poss ib le examples fo r a p a r t i c u l a r f i rm inc lude an increase in f i rm-spec i f i c human cap i t a l or maintenance fo r machines. To become the 'produced' input , the input must be " . . . i n s t a l l e d and adapted by an appropriate app l i ca t i on of the f i r m ' s resources " . This i n te rp re ta t i on emphasizes tha t , s ince production o f output and changing input l e ve l s are j o i n t processes, a more rap id change in input l eve l s can be obtained only a t the expense of output ( i f the resources are g iven ) , or by increas ing resources ( i f output i s g iven ) , each r e su l t i ng in increased interna l costs of adjustment. However, d iminishing marginal costs could a r i se due to , perhaps, g i n d i v i s i b i l i t i e s , as in increas ing returns to t r a i n i ng programs. Another c l o s e l y re la ted ra t iona le for the assumption o f convexity o f adjustment costs i s that i t i s more expensive to do things qu ick ly than s lowly , i . e , "haste makes waste". T h i s , however, may be true only to a degree; i t may be cheaper to expand the p lant s ize over a year rather than a month or a day, but i t i s not necessar i l y less cos t l y i f expansion occurs over ten y ea r s . Hence, i t may be the case emp i r i c a l l y that concav i ty of adjustment costs occurs to a point and then convexity obta ins . E s s e n t i a l l y t h i s appears to be an empir ica l i s sue . An impl i ca t ion o f th i s d iscuss ion that has been examined in the l i t e r a t u r e i s that increas ing marginal costs o f net investment may be more reasonable -16-with interna l than external cos t s . No new adaptation by the f i rm i s required fo r replacement. However such a demand fo r new cap i t a l s t i l l puts the same pressure on external costs in c ap i t a l markets. Thus, s ince to ta l cos ts o f adjustment conta in both in terna l and external cos t components, to ta l Q adjustment costs might best be spec i f i ed as a funct ion o f gross investment. The question o f concavi ty versus convexity i s very important. S p e c i f i c a l l y , the f i rm w i l l not adjust i t s cap i t a l stock slowly i f costs of adjustment are l i nea r or concave^ , but rather w i l l respond to a change in market cond i t ions by ad just ing f u l l y in the f i r s t pe r iod . I f costs are convex, however, the f irm w i l l d i s t r i b u t e i t s response over time as i s s p e c i f i e d in the f l e x i b l e acce le ra tor mechanism. Unfortunate ly , i t i s not i n t u i t i v e l y obvious whether tota l adjustment costs are convex, or which component determines the convex i ty , p a r t i c u l a r l y s ince there are many d i f f e r e n t types o f costs that may be r e f l e c t ed in these very general cost funct ion s p e c i f i c a t i o n s . I t therefore becomes an empir ica l question to determine the shape o f the adjustment costs funct ion as well as the quant i ta t i ve importance of the d i f f e r e n t components. It i s important to cons ider a l t e rna t i ve explanat ions for slow adjustment. The de l i ve ry lag approach emphasizes the " instantaneous" component of the " c o s t l e s s and instantaneous adjustment" assumption. A lag becomes a cons t r a i n t , however, only in the case of uncerta inty about the fu tu re , fo r otherwise the f irm j u s t orders ahead appropr ia te ly to ensure i n s t a l l a t i o n i s completed at the des i red t ime. I f the f i rm faces uncerta in de l i v e r y l ags , i t must try to an t i c ipa te future events. Another a l t e rna t i ve explanat ion could be a combination of uncerta inty about the future and i r r e v e r s i b i l i t y o f investment dec i s ions , i . e . , Z>0. Th is could be in te rpre ted as the consequence of substant ia l adjustment costs assoc iated with the sa le or l eas ing of o ld cap i t a l goods due to the absence of a good second hand or l eas ing market. -17-The only time t h i s cons t ra in t would be b inding would be i f the der ived demand for c ap i t a l were decreasing more r ap id l y than cap i t a l were deprec i a t ing . Th is reasoning impl ies that demand f l u c t u a t i o n s , and therefore imperfect ly compet i t ive aspects of the production process , must be recognized as important components of the f i rm ' s behavior. Stephen Nickel 1 (1978) has compared the consequences of these a l t e rna t i ve sets of assumptions on the investment patterns of a f i rm faced with an expected product demand increase , to the case o f increas ing marginal costs of adjustment where the f i rm an t i c ipa tes the demand increase by r a i s i n g i t s cur rent rate of investment. The r esu l t s in the d i f f e r e n t cases are s im i l a r in terms of a "smoothing" e f f e c t on the demand fo r capac i t y , but Nickel 1 argues that lags or i r r e v e r s i b i l i t y and uncerta inty are more p laus ib le a p r i o r i . He concludes that costs o f adjustment may be important for slow adjustment, although they may not be convex. I f they are not, they may cause a gap between the cos t of cap i t a l and i t s marginal revenue product , but w i l l not necessa r i l y r e su l t in the emp i r i c a l l y observed phenomenon o f slow and caut ious responses of investment dec i s ions . In many recent models, slow and caut ious investment occurs because o f costs of adjustment. These are modelled by spec i f y ing increas ing marginal in terna l costs on net investment for quas i- f ixed inputs , represented by a production funct ion of the form: 1.2.5) Y = F ( x , x , v , t ) , such that i f absolute l e ve l s o f the quas i- f ixed fac tors vary ( x ^0), output f a l l s for any given amounts of x and v because of the necess i ty to devote resources to changing the stock. Thus the assumption i s s u f f i c i e n t l y abst rac t and general to encompass var ious forms of convex adjustment c o s t s . -18-Costs of adjustment can be e x p l i c i t l y incorporated into the opt imiz ing process of the f i rm by employing the dual r e l a t i on and by assuming that f i rms minimize normalized var iab le costs condi t iona l on output (Y), l eve l s of and changes in the quas i-f ixed inputs (x and x ) , and the pr i ces of the var iab le inputs , P.. Such a r e l a t i on i s character ized by the dual normalized r e s t r i c t e d cos t f unc t i on : where P.=P./P,=the pr i ce of the j t h var iab le input normalized by the U J ^ pr i ce of the f i r s t va r i ab le input . The long run economic problem of the f i rm i s then to minimize the present value of the future stream of c o s t s : where z . i s gross investment in x^ and a.=u./(r+d.)=the normalized asset p r i ce of x^, where un- i s the normalized rental p r i ce and d. i s the deprec ia t ion r a t e . This i s solved by choosing the time paths of the contro l v a r i ab l e s , v ( t ) , x ( t ) , and the state var iab le x(t ) that minimize L (0 ) , given x(0) and v ( t ) , x(t )>0. The so lu t ion can be accomplished in two steps. F i r s t , the normalized r e s t r i c t e d var iab le cos t funct ion G i s derived as above, incorporat ing the short run determination of the optimal demand for var iab le fac tors cond i t iona l on the values of the quas i-f ixed f a c t o r s . The short run so lu t ion can be represented by a var iab le input opt imizat ion cond i t ion analogous to (1 .2 .3 ) , but in a cos t minimizat ion instead of p r o f i t maximization context , and has been expressed as Shephard's lemma: 1.2.6) 1.2.8) 3G/3W, = v. = the short run optimal demand fo r the var iab le input v . . -19-To determine the optimal demand for cap i ta l-- to optimize over the quas i- f ixed inputs—one must cons ider the en t i r e optimal accumulation path over time s ince costs of adjustment make th i s problem t r u l y dynamic. Thus i t i s more complicated to der ive an equation l i k e (1 .2 .4 ) . Now one charac te r izes the long run "des i red " cap i t a l stock analogously to (1.2.4) as a marginal equa l i t y , and then determines the f i r m ' s optimal investment path to reach t h i s po in t . When the var iab le cost funct ion G i s subst i tuted into (1.2.7) and the r e su l t i ng funct ion i s integrated by pa r t s , one obta ins : poo 1.2.9) L(0) + ^ 1 - a i x . ( 0 ) =1 e " r t (G (w ,x , x , Y , t ) + ^ . u . x i ) d t . The so lu t ion to t h i s opt imizat ion problem has a s t ra ight forward i n t e rp r e t a t i on . Since G assumes short run opt imizat ion behavior with respect to the va r i ab le inputs cond i t iona l on Y ( t ) , w(t) , x ( t ) , and x ( t ) , the opt imizat ion problem (1.2.7) fac ing the f irm i s to f i nd among a l l the poss ib le combinations given by the va r i ab le cost funct ion (1.2.6) that time path of x ( t ) , x ( t ) , which minimizes the present value of a l l co s t s . The der i va t ion of the so lu t ion to (1.2.7) i s the second step o f the dynamic opt imizat ion problem. The necessary condi t ions can be obtained using e i t he r the Euler f i r s t order condi t ions or the maximum p r i n c i p l e . The maximum p r i n c i p l e y i e l d s a system of 21 f i r s t order d i f f e r e n t i a l equations which can be manipulated to y i e l d the I second order d i f f e r e n t i a l equations cha rac te r i z ing the so lu t ion to the Euler equat ion. Assuming s t a t i c expectat ions with respect to normalized fac to r pr ices and output, one can write the Hamiltonian for the maximum p r i n c i p l e as : 1.2.10) H (x ,x ,n , t ) = e " r t (G (w ,x , x , Y , t ) + S i u i x i ) + where n i s a vector of co-state v a r i ab l e s . From the necessary c o n d i t i o n s , -20-th i s imp l i es : 1.2.11) -G x - r G x - u + G^x + G^x = 0, where the subscr ip ts denote der i va t i ves and x* i s the second pa r t i a l de r i va t i ve with respect to t ime. The steady state (long run) so lu t ion at x=x=0 s a t i s f i e s : 1.2.12) -G x(w,x*) = u + rG x (w ,x* ) , and can be in terpre ted as fo l lows : the left-hand side i s the marginal benef i t to the f i rm of changing the quas i-f ixed inputs (the decrease i n var iab le costs from purchasing cap i t a l equipment or h i r i ng addi t iona l s k i l l e d workers) , while the right-hand s ide i s the marginal cos t (user cost plus amortized marginal adjustment cost ) of a change in the amount of cap i t a l or s k i l l e d labor serv ices at x=0. Thus, t h i s i s a f am i l i a r marginal benef i t equals marginal cos t condi t ion where the marginal adjustment cost dr ives a wedge between the usual s t a t i c marginal benef i t equals user cost cond i t ion given by (1 .2 .4 ) . Arthur Treadway (1969, 1971, 1974) has derived a pa r t i a l adjustment or f l e x i b l e acce le ra tor r e l a t i on fo r the quas i-f ixed fac tors of th i s model, which expresses the optimal path toward the long run optimal level of c a p i t a l . Th is f i n a l step along with (1.2.8) determines within an e x p l i c i t dynamic opt imizat ion process a completely integrated s p e c i f i c a t i o n of short run var iab le input demands and quas i-f ixed input accumulation. S p e c i f i c a l l y , Treadway demonstrated that an approximate so lu t ion fo r the homogeneous d i f f e r e n t i a l equation system (1.2.11) can be generated from the roots of the corresponding quadratic form: 1.2.13) - G * X X M * 2 - rG * x .M* + G * x x + r G * x x = 0. Subst i tu t ing th i s into the usual so lu t ion to a second order d i f f e r e n t i a l equat ion, r esu l t s i n : -21-X-it X ? t 1.2.14) x(t) - x* = A : e + A 2 e c , where the I x^'s and I x 2 ' s are symmetric around r/2 (Treadway (1971)). Note that in order fo r t h i s to converge to x* (the s tat ionary point charac te r ized by (1.2.12)) as t>«> , the pos i t i v e roots Xj must have no i n f l uence . Hence, with A^=0, and a f t e r ar ranging, one obta ins : x ? t 1.2.15) x* - x(t) = -A 2 e , or 1.2.15' ) x = x 2 ( x * - x ( t ) ) , which i s the f l e x i b l e acce le ra tor form for investment in quas i- f ixed fac tors g der ived endogenously with in a dynamic opt imizat ion process. In t h i s model adjustment costs cause the net benef i ts of i n s t a l l i n g an extra un i t of c ap i t a l stock to be a funct ion of the complete time path of c a p i t a l . Thus adjustment costs d i s t i ngu i sh t h i s dynamic model from the analogous model without adjustment c o s t s , which as we have seen i s e s s e n t i a l l y s t a t i c . However, numerous r e s t r i c t i v e assumptions are s t i l l inherent in t h i s dynamic model. For example, the s i m p l i c i t y of the so lu t ion depends c r i t i c a l l y on the assumption of s t a t i c expectat ions on output l e ve l s and r e l a t i v e p r i c e s . I f expectat ions were non-sta t i c , adjustment of x would depend on the en t i r e time path o f expected future output and p r i c e s , which w i l l imply cons iderably more complexity. For example, with one quas i- f ixed i n p u t — c a p i t a l , s t a t i c expectat ions on exogenous var iab les are required to determine a "des i r ed " cap i t a l s tock, K*, to generate the f l e x i b l e acce le ra tor form. Th is assumption i s f a r too r e s t r i c t i v e in a world of technologica l change. Moreover, in r e a l i t y , output demand i s constant ly changing and not only absolute p r i c e s , but a lso r e l a t i v e -22-pr i ces vary over time. Thus, even i f one assumed ce r t a in expectat ions, one would need some assumptions concerning the way exogenous var iab les change over 12 t ime. Nickel 1 (1977) and others have found that in general when exogenous var iab les are expected to be changing, instead of aiming at an equ i l ib r ium leve l of cap i t a l the f i rm aims at an equ i l ib r ium cap i ta l stock based on current pr ices plus a weighted sum of the d i f fe rences between th i s equ i l ib r ium cap i t a l stock and the d i f f e r e n t equ i l ib r ium cap i t a l stocks corresponding to the l eve l s of demand and pr i ces expected fo r a l l future per iods . Nickel 1 fur ther emphasizes tha t , unl ike the f l e x i b l e acce le ra tor model with constant p r i c e s , often no unique form of the investment path can be derived with var iab le p r i c e s . However, given c e r t a i n t y , some sort of complex dependence of present dec is ions on future exogenous var iab les w i l l e x i s t , whereas with uncerta inty i t i s in general not even poss ib le to charac ter ize the optimal paths a n a l y t i c a l l y . Moreover, with uncertainty i t i s a lso d i f f i c u l t to def ine the object ive func t ion , as net income i s a random var iab le so that the goal of 13 maximizing present value may not be app l i cab le . Thus, although both the s t a t i c and ce r t a in expectat ions assumptions are too r e s t r i c t i v e and d i f f i c u l t to r e l ax , the ce r ta in ty assumption presents the most ser ious d i f f i c u l t i e s . Another important r e s t r i c t i o n on the behavior of the f i rm in the basic dynamic model presented above i s the assumption of pr ice-tak ing in both the input and output markets. For example, i t has already been noted above that some degree of monopsony in the cap i t a l market i s required in order to have external costs of adjustment. In add i t i on , output i s inc luded in the investment f unc t i on , which i s cons is tent with the idea in s t a t i c models that output i s exogenous to the f i r m . However, as investment i s dynamic, i t i s not c l e a r that t h i s i s v a l i d ; in a short run sense the f i rm i s not pe r fec t l y 14 competi t ive . F luc tuat ions in output demand are important determinants of investment with slow adjustment response. Imperfect competit ion in the output -23-market (monopol ist ic elements) impl ies that the f i rm maximizes present value given that output i s adjusted according to demand. When there are no inventor ies in the model, the f i rm must vary the p r i ce o f output (p) cont inuously so that t h i s i s t rue . Th is i m p l i c i t assumption inherent in the imperfect competit ion case creates d i f f i c u l t i e s , as output pr i ce does not a c tua l l y exh ib i t substant ia l f l e x i b i l i t y . However the e f f e c t of f l uc tua t ions in demand on the f irm in terms o f monopol ist ic elements i n the f i r m ' s behavioral pa t te rns , seem more important to recognize. In add i t i on , i f va r i a t ions in cap i t a l (or capac i ty ) u t i l i z a t i o n by the f i rm are e x p l i c i t l y incorporated as mechanisms to keep demand in l i n e with supply , the imperfect ly competit ive model i s more r e a l i s t i c . Output p r i ce does not need to be as f l e x i b l e i f va r i a t ions in demand can be accommodated by varying other f l e x i b l e va r iab les such as labor 15 and/or the rate o f c ap i t a l u t i l i z a t i o n . Another assumption to cons ider—tha t of constant exponential deprec ia t ion—ignores the dependence o f deprec ia t ion on maintenance expenditure, i n t ens i t y of use, and pr i ces of va r iab le i n p u t s ' ^ . There are two d i s t i n c t aspects to th i s assumption. F i r s t one could cons ider non-exponential decay, which impl ies that the age s t ruc ture of the c ap i t a l stock matters ; the comparative value to the f i rm of o ld goods as opposed to new ones would depend on external economic c ircumstances. Th is i s p a r t i c u l a r l y important fo r replacement investment. However, i f the age s t ructure of cap i t a l remained constant over t ime, the potent ia l replacement-to-capital r a t i o would s t i l l be constant . I t may be even more important to cons ider the ways in which the f i rm can in f luence the rate of deprec i a t ion , p a r t i c u l a r l y through maintenance expenditure and i n t ens i t y of use. Maintenance expenditures would not seem to in f luence the ana lys is p a r t i c u l a r l y , but only provide a way to increase -24-revenue with a t rade-of f between maintenance costs and cap i t a l decay such that on the optimum path the costs balance at the margin. However, i f the i n t ens i t y o f use o f c ap i t a l goods ( cap i ta l u t i l i z a t i o n ) a f f e c t s the rate of decay, i t therefore imposes a "user cos t " of using cap i t a l goods. This was recognized by Keynes as a l i nk between present and future dec is ion making; c ap i t a l u t i l i z a t i o n i s a dec i s ion va r i ab le fo r the f i rm which var ies with output and input p r i c e s . It i s thus an important cons iderat ion a f f e c t i ng investment dec i s ions . F i n a l l y , other dec i s ion va r i ab les such as inventory accumulation could be incorporated in to the ana lys i s as " bu f f e r s " fo r f l uc tua t ions in the economy. I f output inventor ies were inc luded , fo r example, the f irm would not be constra ined to equate output and sales in each time pe r iod . Th is would provide an addi t iona l choice va r i ab le with which to adjust smoothly to economic shocks, p a r t i c u l a r l y c y c l i c a l v a r i a t i ons . To summarize, in t h i s sect ion I have discussed genera l iza t ions which would r e s u l t in a r i che r set o f choices ava i l ab l e to the f i rm and increase the r e a l i t y o f previous models, but would a l so increase the complexity of the a n a l y s i s . Therefore , in s e t t i ng up a model for empir ica l implementation, a t rade-of f ex i s t s between real ism and t r a c t a b i l i t y . The new model I present in the next sec t ion attempts to incorporate some but not a l l of the more important aspects of the genera l iza t ions discussed above. III. The Monopoly Model -25-In th i s sect ion I extend the recent l i t e r a t u r e on emp i r i ca l l y implementable dynamic fac to r demand model l ing to a more general monopoly framework. S p e c i f i c a l l y , I permit output p r i ce and quant i ty to be v a r i ab l e , and spec i fy both in terna l and external adjustment costs to be a funct ion o f gross rather than net investment, but re ta in here the assumption of myopic expecta t ions . Th is expectat ions assumption w i l l be genera l ized in Essay 2. The r e s u l t i n g framework incorporates some of the genera l iza t ions discussed in the previous s e c t i on , ye t keeps the model a n a l y t i c a l l y and emp i r i c a l l y t r a c t a b l e . Let the f i rm ' s production p o s s i b i l i t i e s be charac ter ized by: 1.3.1) Y t = F ( v t , x t , z t , t ) , where i s the f i rm ' s output, z .^ = x t + d x t , and F s a t i s f i e s the usual r egu l a r i t y condi t ions for a production func t ion . Assume a lso that costs of adjustment inc lude both external and in terna l costs of adjustment on gross investment;*^ to t a l adjustment costs may be convex even i f one component i s not . The s p e c i f i c a t i o n presented here i s thus f a i r l y genera l , and allows one to t e s t for the more r e s t r i c t i v e spec i f i c a t i ons embodied in previous s t u d i e s — s o l e l y in te rna l or external adjustment c o s t s , depending e i t he r on net or gross investment. Some ind i ca t i on of whether the convexity assumption i s v a l i d can a l so be obtained by observing the signs and magnitudes of the estimated cos t parameters. Internal costs are def ined by the r e l a t i o n : 1.3.2) 3Y/3Z t <0, -26-i . e . , the in terna l costs of a change in the stock of the quas i-f ixed fac tors are represented in terms of foregone output. Purchase costs of asset accumulation and costs external to production a c t i v i t y can be represented in aggregate as : 1.3.3) C^z^ ) = a ^ . + a ^ f z j ) , where S. i s the external cost of adjustment funct ion and the 'ha ts ' represent non-normalized p r i c e s . Although 0^(0) = 0, in a steady state there i s pos i t i ve replacement investment and therefore pos i t i ve gross investment equal to d . x . . Thus, instead of 0^(0) as with net investment in the steady s ta te , one now obta ins : 1.3.4) C j f d ^ j ) = a 1 d 1 x 1 + a ^ . ( d ^ . ) , where C . ' (z. )>0, C. "{z.)>0, 1=1 , . . . I . Note that external costs of accumulation of quas i-f ixed fac tors do not a f f e c t current product ion, fo r these costs are not included in the production funct ion (1 .3 .1 ) . Except fo r the monopsony power inherent in the assumption of external costs of adjustment fo r the quas i-f ixed inputs , input pr i ces are assumed f i xed and exogenous. Thus I assume (P j ) > u j "is the constant exogenous vector of va r i ab le input pr i ces which could be a r e s u l t of competit ion i n the va r i ab le- fac to r markets. Complications invo lv ing endogenous deprec ia t ion and the existence of output inventor ies are a lso ignored here. I assume the f i rm maximizes net revenues or cash flows subject to a downward s loping demand curve fo r i t s product. In t h i s sense i t can be considered to possess some "monopoly" power. The cos t minimizing case analyzed e a r l i e r in th i s essay i s simply a r e s t r i c t e d form of the net revenue -27-maximizing case, given the cons t r a in t that output i s exogenous (Lau (1976)). Although the monopol ist ic element i s p a r t i c u l a r l y s i g n i f i c a n t at the industry leve l in the empir ica l ana l y s i s , even at the leve l of the f i rm i t i s important to al low fo r imperfect ly compet i t ive aspects of market behavior , rather than simply assume output to be exogenous. Marc Nerlove out l ined the reasons fo r t h i s as fo l lows : "Investment i s a dynamic phenomenon, and in a dynamic context i t simply does not make sense to t rea t output as exogenous to the business sec to r . . . . Th is means, at l e as t in some short-run sense, that the f i rm cannot be considered pe r f e c t l y compet i t i ve . . . . Many f irms are perhaps r e a l i s t i c a l l y viewed as ad just ing quant i ty to changes in demand most frequent ly but a lso ad jus t ing pr i ces more i n f r equen t l y . " (Nerlove (1972), p. 242) At l eas t four approaches fo r se t t i ng up the p r o f i t maximizing problem of the monopol ist ic f i rm are a v a i l a b l e . These have been discussed by Diewert (1978). Two appear to be p a r t i c u l a r l y promising. The f i r s t approach i s to def ine a va r i ab le p r o f i t funct ion as a funct ion of va r iab le input p r i c e s , quas i- f ixed input l e v e l s , and the shadow or marginal p r i ce of output as def ined by the inverse demand func t i on . Th is approach i s a t t r a c t i v e , fo r the va r i ab le input demand funct ions are symmetric; one input does not have to be chosen to be the numeraire. However, i t requires knowledge of the monopol is t ' s demand curve and a lso r e su l t s in a system of demand equations which i s simultaneous and not r ecu rs i ve . I therefore use an a l t e rna t i v e approach, based on the representat ion of the f i rm as an imperfect competitor f ac ing the inverse demand func t ion : 1.3.5) p = e + D(Y), where p i s the p r i ce o f output , and e represents the in f luence on demand of other exogenous v a r i a b l e s — f o r example disposable income for the per iod i f the monopolist were s e l l i n g to consumers, or a l i n ea r homogeneous funct ion of the -28-18 p r i ces other producers face i f the monopolist were s e l l i n g to producers. Note that (1.3.5) i s a stra ightforward genera l iza t ion of the pe r fec t l y competi t ive case , which can be tested as a specia l case. Now the f i rm ' s intertemporal p r o f i t maximization problem i s : 1.3.6) Max V(0) = jQ e " r t R(t) d t , where: 1.3.7) R(t) = p t Y t - E j P j t V j t -E1-C1-(zit), subject to : 1.3.7' ) p t = e + D(Y t ) Y t = F ( v t ' x t ' z t } L1 c^z.^ = £ . a 1 ts.(z u) z.4. = + d . . x . . , = gross investment, i t i t 11 i,t-1 3 The f i r m ' s dec is ion problem at time zero i s therefore to choose v, x, and p (which of course, impl ies Y by the inverse demand funct ion re l a t ionsh ip ) to maximize the present value of the cash flow R(t) (or net r e c e i p t s ) . This opt imizat ion problem i s solved in three steps: ( i ) f i r s t var iab le costs are minimized, given x, z , and Y; ( i i ) then, given th i s va r iab le cos t f unc t i on , the maximum (instantaneous) var iab le p r o f i t obtainable at time t i s der ived , condi t iona l on the l eve l s of x and z ; and, f i n a l l y , ( i i i ) the present 19 value of to ta l net rece ipts are optimized over the quas i-f ixed inputs . By decomposing the problem in th i s way I i m p l i c i t l y make some r e s t r i c t i v e assumptions. In p a r t i c u l a r , the pr i ce of output i s assumed here to be instantaneously and c o s t l e s s l y v a r i ab l e , j u s t as are the quant i t i es of the var iab le f a c t o r s . The f ac t that the instantaneous p r o f i t maximization over output becomes a s t a t i c problem rather than an intertemporal problem depends on t h i s r e s t r i c t i o n , plus the assumption that the pos i t i on of the demand curve a t time t i s not a f fec ted by the opt imizing dec is ions (such as the pr i ce of output) of the f i rm in any previous yea r s , and that the short run and long run -29-output demand func t ions , and therefore output e l a s t i c i t i e s of demand, are the 20 same . Although the assumption of per fec t f l e x i b i l i t y of p i s not very r e a l i s t i c , the f i rm enjoys a l t e rna t i ve poss ib le adjustment cho ices , such as varying capac i ty (and/or cap i t a l ) u t i l i z a t i o n over demand c y c l e s , r esu l t i ng in a 21 smoothing of p r i ce f l uc tua t ions . One could therefore consider adjustments in var iab le inputs , such as energy or l abor , to be adjustments in capac i ty u t i l i z a t i o n . This captures at l eas t some of the poss ib le a l t e rna t i ves to w i ld l y f l u c tua t ing output p r i c e s , which in r e a l i t y are not observed. C lea r l y there are costs of adjust ing output p r i ce which could very p laus ib l y be increas ing at the margin, analogous to costs of adjustment for the quas i- f ixed 22 inputs . I now b r i e f l y analyze each of the three steps in turn to der ive the system of behavioral equations f o r the f i rm . i ) Cost Minimizat ion Given Y t , P-t, x.^, and x ^ , the f i rm i s assumed to minimize: 1.3.8) L(0) = jQ e " r t fyj)dt' subject to the production funct ion cons t r a in t . A fac to r requirements funct ion fo r the var iab le input , v^(t ) , can be obtained by inver t ing the production funct ion to ob ta in : 1.3.9) v 1 ( t ) = F _ 1 ( v 2 ( t ) , . . . , V j ( t ) ; x t , z t , Y t ) . Subs t i tu t ing th i s in to L (0 ) , and so lv ing the opt imizat ion problem fo r the var iab le inputs v . ( t ) , one obtains the cost minimizing var iab le f ac to r J demands cond i t iona l on x^, z^, and Y^. The opt imizat ion problem can then be wr i t ten as : 1.3.10) min v L(0) = minj'Q e~rZ ( P 1 F _ A ( v 2 ( t ) , . . . V j ( t ) ; x ,z ,Y ) + £ P j V j ) . The f i r s t order condi t ions fo r th i s minimizat ion problem are : 1.3.11) F"vjl = -P. , where P j - P./P,. Note that t h i s impl ies F~J<0, so that an increase in the a v a i l a b i l i t y of the j va r iab le f ac to r reduces the amount of v^ required to produce a 23 given leve l of output. Equation (1.3.10) can be solved f o r the cos t minimizing short run var i ab le f ac to r demand funct ions v ( t ) , thus r e su l t i ng in a normalized r e s t r i c t e d cos t func t ion : 1.3.12) G t = E P j t v j t = G ( P t , x t , z t , Y t ) . 6 i s the minimum var iab le cost of producing Y t at time t , cond i t iona l on x, z, P, and Y, and s a t i s f i e s proper t ies r e su l t i ng from the regu la r i t y cond i t ions 24 on the production func t i on . These have been derived in Lau (1976) . For my purposes here, two of these propert ies are p a r t i c u l a r l y important: 1.3.13) sG/sP. = v . , J J the cond i t iona l cost minimizing input l e v e l , and, 1.3.14) 3G/3X.. = - u . , the negative of the normalized user cos t or shadow pr i ce of the serv ice flow from the stock of x . . Equation (1.3.13) i s analogous to Shephard's Lemma and equation (1.3.14) expresses the shadow pr i ce of the f i xed f ac to r cons t r a in t . Since G charac ter izes the cost minimizing demand funct ions fo r var iab le inputs cond i t iona l on x, z , and Y, one can now subst i tu te back in to the •30--r t ,--1, -31-object ive funct ion (1.3.6) and perform the second step of the opt imizat ion problem. Th is i s accomplished as fo l lows . i i ) P r o f i t Maximization The f i rm now wants to maximize normalized var iab le p r o f i t s at time t : 1.3.15) maxY R ' ( t ) = ((e+D(Y))Y - G (P ,x ,z ,Y ) ) . The so lu t ion to th i s opt imizat ion problem can be character ized by: 1.3.16) (e+D(Y)) + D'(Y)Y - (aG(P,x,z,Y)/3Y) = 0, A A or noting that p=(e+D(Y))=p/P^, we have: 1.3.17) p = -D'(Y)Y + (sG/sY). This so lu t ion along with those of the v- imbedded in the cos t funct ion (from 1.3.13) together charac ter ize the maximum normalized var iab le p r o f i t s obtainable at time t , condi t iona l on the leve l s of x and z . Note that the shape of the demand curve does not have to be known to estimate th i s model; ra ther , the parameters determining the shape of the demand funct ion can be estimated simultaneously using conventional econometric techniques. C a l l i n g t h i s "pseudo-variable p r o f i t func t ion" H (P ,x ,z ) , I subst i tu te back in to the o r i g i na l object ive funct ion (1.3.6) to obtain the funct ion expressing the over-al l maximization problem fac ing the f i r m . i i i ) Intertemporal maximization over x , z . The f i rm now wants to choose x and z to maximize the present value o f : 1.3.18) R(t) = H(P,x,z) - (K-a.z. +E 1- a 1-VV ) ) = H(P,x,z) - ( £ j a j X j +E1-a1.xid1. -EjajSj(x^ .x.)), -32-where the terms in the parentheses represent the to ta l purchase plus external costs of adjust ing the cap i t a l stock. This maximization problem can be rewri t ten as : 1.3.19) m a x X ) Z V(0) = JQ e " r t (H(w,x,z) - ^ . a . d . x . -0 e'^E-a^. d t , where the l a s t part of the expression can be integrated by parts and subst i tu ted back into V(0) to y i e l d : •r> 1.3.20) X)n-Vi ( 0 ) +^ { 0 ) = /o [ e _ r t ( H { t ) ~^iuixi -Ei<liVx1»*i))] d t , 0 W(0) d t , where u^=a.(r+d.) i s the normalized user cos t associated with the serv ice flow from x. when marginal costs of adjustment are constant , and a., i s the asset p r i c e . It i s c l e a r that fo r each t the so lu t ion to th i s maximization problem depends on a l l future expected p r i c e s . Here I re ta in the e a r l i e r assumption of constant r e l a t i v e p r i ce expectat ions ( s t a t i c expecta t ions ) ; in t h i s sense the f i rm i s myopic in i t s expectat ions . Such an assumption i s not completely unreasonable fo r p r i c e s , as they are expressed in r e l a t i v e terms. For expectat ions on output quant i ty , which before were a lso assumed to be myopic, here with the behavioral assumptions the f i rm i s able to choose i t s output l e v e l , subject to the demand curve. Thus the expectat ion assumption here i s not as r e s t r i c t i v e as i t was in the cos t funct ion case , in which the output level was assumed to be exogenous. In a sense, the re levant myopic -33-expectat ions assumption in th i s monopoly model becomes that of assuming a s tat ionary output demand curve over t ime. Note that t h i s expectat ions assumption does not reduce the problem to a maximization of instantaneous p r o f i t . The optimal cap i ta l stock at time t i s dependent on the stock at time t-1 , so the problem remains intertemporal even 25 though only current values of var iab les are used in dec is ion making. Maximizing (1.3.20) i s equiva lent to maximizing the o r i g i na l V(0) (s ince ^•a-x-fO) i s an i n i t i a l cond i t i on ) . S i m i l a r l y , maximizing V(0) with respect to v, x, and Y i s equiva lent to maximizing V(0) with respect to x since (1.3.20) incorporates the optimal v(t ) and Y(t) cond i t iona l on x ( t ) . The Euler necessary condi t ions fo r th i s problem can be wr i t ten as : 1.3.21) 3W(0)/3Xj - d/dt(dW(0)/3Xj) = 0, o r , in vector form, 1.3.22) H x - aS x - u - H^x - H x x x + a S x x x + r (H x -aS x ) + aS x x x = 0. In the s tat ionary state (at x* (p,P ,r ) obtained by se t t ing x-=x.=0), one obta ins : 1.3.23) H x (P ,x* ,x ,Y) - aS x (x* ,x ) + r ( H x ( 0 ) - a S x ( 0 ) ) - u = 0. H(P,x,z) has been def ined above as the maximum var iab le p r o f i t obtainable V at time t condi t iona l on x and z , which can be character ized by: 1.3.24) H = e+D(Y)Y - G(P,x,z,Y) = e+D(Y)Y - G (P ,x ,x ,Y ) . Thus I now have: 1.3.25) rl (P ,x*,x,Y) = -G ¥ (P ,x* ,x ,Y ) A A H x (P ,x* ,x ,Y) = -G . (P ,x* ,x ,Y ) , -34-which impl ies that the Eu ler cond i t ion becomes: 1.3.23' ) -G x (P ,x* ,x ,Y ) - aS x (x* ,x ) - r(G*(0) + aS^O)) - u = 0. Th i s equation s tates that the marginal value product o f x i s equal to the marginal accumulation cos t (at the steady s t a t e ) . This i s the usual cond i t ion of equa l i t y between marginal benef i t s and marginal cos t s , which would be more c l e a r l y in te rpre ted i f (1 .3 .23 ' ) were rewri t ten as: 1.3.23") -G K - aS K = r(G£+aS£) + u. Thus, the f u l l benef i t s r e su l t i ng from an addi t iona l un i t of the cap i t a l s tock , or marginal e f f i c i e n c y of cap i t a l (the l e f t hand s ide of the equat ion) , i s set equal to the f u l l costs of obta in ing the incremental un i t , the rental cos t plus the amortized adjustment c o s t s , x* i s therefore the steady state or long run prof i t-maximizing demand for x obtained by so lv ing ( 1 . 3 . 2 3 ' ) . The steady state demand for the va r i ab le inputs , v* ( t ) , i s obtained by subs t i t u t i ng x(t )=x*(t) in the va r i ab le input demand equations charac te r ized by (1 .3 .13 ) . As an a l t e rna t i v e representa t ion , note that t h i s cond i t ion impl ies that the simple cha rac te r i za t i on of the shadow pr i ce o f cap i t a l from the normalized va r i ab le cos t funct ion as represented by 1.3.14 i s too simple in the f u l l dynamic s p e c i f i c a t i o n . The value o f inc reas ing the c ap i t a l s tock, given by - G K , must be augmented by the value of the saved external costs represented by -S K , and a lso by the amortized adjustment costs charac te r ized by -rG^ and -raS^, r e spec t i v e l y . Thus the s p e c i f i c a t i o n o f the f i r m ' s problem as a f u l l y dynamic problem, inc lud ing not only the stock value (marginal e f f i c i e n c y o f c a p i t a l , MEC) but the costs of a c tua l l y ad just ing the stock (marginal e f f i c i e n c y o f investment, MEI), r esu l t s in a more complex s p e c i f i c a t i o n of the shadow value of c a p i t a l . -35-As before , the optimal path of x can be represented as an approximate so lu t ion in the neighborhood of x* to the l i nea r d i f f e r e n t i a l system given by the mul t i va r i a te f l e x i b l e acce lera tor of the form: 1.3.26) x = L*(x* - x ) , o r , 1.3.26') z = k + dx = L*(x* - x) + dx, where L* i s the matrix of endogenous adjustment parameters der ived from the stable roots s a t i s f y i ng the cond i t i on : 1.3.27) -(G*• •+aS*< • )L* 2 - r ( G * " + a S * - « )L* + (G* +aS* +rG*. +raS*- ), XX XX XX XX XX XX XX XX ' where * ind ica tes eva luat ion at x*, and where G*« i s assumed to be A A symmetric . With only one quas i-f ixed input , c a p i t a l , L* can be wri t ten as : 1.3.28) | A | = - .5 ( r- ( r 2 + 4 ( (G* x x +aS* x x +rG* x x +raS* x x ) / (G* x ^+aS*^0 ) ) * 5 ) . The normalized r e s t r i c t e d cos t funct ion incorporates the production funct ion and therefore embodies parameters of the technology. With three var iab le inputs ( labor , L, energy, E, non-energy intermediate mate r i a l s , M) and one quas i- f ixed input ( c a p i t a l , K) , gross investment in the quas i- f ixed input i s Z=K+dK. The normalized r e s t r i c t e d cos t funct ion i s of the form G=G(Y,t,PE-,P|V],K,Z) which can be approximated by the fo l lowing va r i a t i on of 27 a quadrat ic form : 1.3.29) G = L + P EE + PM = Y (a n + a^ t + a £ P E + + a yY+ y ^ P ^ + - 5 ( ^EE P E V M 1 + V E Y + V M Y + ttV + ^ M t V + aKK + a zZ + . 5 ( Y K K K 2 + Y Z Z Z 2 ) + y ^ K + y ^ + c ^ K t + Y y K YK + ^ EZ P E Z + W + a Z t Z t + Y y Z Y Z + Y K Z K Z -- 3 6 -Using the property that sG/sP. = v. (j=E,M), one can obtain short run cost-minimizing quant i t i es demanded f o r E and M. One can a lso use the f a c t that G=L+PEE+PMM, so that I^G-P^E-P^M, to derive a demand equation fo r The short run var iab le input demand funct ions are ca l cu l a ted a s : 1 . 3 . 3 0 ) E = 3 G / 3 P E = Y( a E + y^Py, + Y ^ + Y E yY + a £ t t ) + y^K + y ^ Z 1 . 3 . 3 1 ) M = 3 G / 3 P M = Y( a M + y E M P E + y ^ + y^Y + a ^ t ) * Y ^ K + y ^ Z 1 . 3 . 3 2 ) L = G-PEE-PMM = Y ( a Q + a ^ t + a y Y - y ^ P ^ - . S f y^P* + Y MM P M ) + a K K + a Z Z + - 5 ( Y K K k 2 + Y Z Z z 2 ) + a K t K t + Y y K Y K + a Z t Z t + Y y Z Y Z + Y K Z K Z ' The fourth equation in the system represents the so lut ion to the p r o f i t maximization problem: 1 . 3 . 1 7 ) p = -D'(Y)Y + (3G/3Y) . In order to der ive D' (Y) , some assumptions must be made regarding the inverse demand funct ion D(Y). I assume th i s can be adequately approximated over the re levant range o f Y by the func t i on : 1 . 3 . 3 3 ) P = D(Y) = aj_ - B x Y , where a^>0, B^>0, are constants . Subst i tu t ing th i s into ( 1 . 3 . 1 7 ) , and a lso subs t i tu t ing fo r sG/sY y i e l d s the output p r i ce equat ion: 1 . 3 . 3 4 ) p = ( e i + 2 a y ) Y + a Q + a ^ t + + + y ^ P ^ + ' 5 ( Y E E P E 2 + YMM PM } + 2 ( Y E y P E Y + Y M y P M Y ) + "EtV + a Mt P M* + Y y K K + Y y Z Z ' -37-A f i f t h equation must be incorporated in to the system to spec i fy e x p l i c i t l y the inverse demand funct ion r e l a t i onsh ip between Y and p and thus to y i e l d a complete system determining a l l re levant parameters. To accomplish t h i s the inverse demand funct ion p=e+D(Y) must be estimated d i r e c t l y in terms of Y to i den t i f y Y as an e x p l i c i t funct ion of exogenous va r i ab l e s . D(Y) was spec i f i ed above, so only e must be determined i n order to incorporate th i s equation into the model. Assume e can be simply formulated as a 2 +B 2 X, where X i s a vector of var iab les exogenous to the system but important f o r determining the shape and pos i t i on of the demand curve; in e f f e c t these are " s h i f t v a r i a b l e s " . The corresponding B 2 i s therefore a vector of parameters. Thus the en t i re inverse demand funct ion r e l a t i onsh ip can be formulated as : 1.3.35) p = ct^ - B^ Y + a 2 + B 2X, or p = a - BjY + B 2X, where a i s the composite in te rcept term a^+a.?. This expression can then be inverted in Y to obtain the d i r e c t demand func t ion : 1.3.36) Y = (a / B ^ - (1/B^p + (B 2 /B 1 )X. F i n a l l y , the optimal path fo r K can be der i ved . F i r s t assumptions about the cos t of adjustment funct ion S(K,K) must be s p e c i f i e d . I assume th i s can be approximated by a quadratic form: 1.3.37) S(Z) = . 5 ( d K K Z 2 ) = . 5 d K K ( K 2 + d 2 K 2 + 2dKK). Using (1 .3 .23 ) , (1 .3 .29 ) , and (1 .3 .35 ) , I then obtain the steady state so lu t ion for K* as: -38-1.3.38) K * = ( ( - 1 / ( Y K K + Y z z ( d 2 + r d ) + Y | < z (2d+r ) + ad K | < (d 2 +rd) ) ) (e^ + Y £ | < P E + YMK PM + "K t* + V + ( d + r ) ( a Z + Y E Z P E + YMZ PM + " Z t * + Y y Z Y ) + u ) ' Given the so lu t ion values fo r (1.3.27) character ized by (1.3.28) and the de r i va t i ves from (1.3.29) and (1 .3 .37) , I subst i tu te in to the f l e x i b l e acce le ra tor form (1.3.26) to obtain the cap i t a l accumulation equat ion: 1.3.39) Z = K + dK = jx | (K*-K_ x) + dK = (- .5( r- ( r 2 + ^ J / y^ + a d ^ ) ) - 5 ) ) ((-1/J) ( a K + Y E K P E + Y M K P m + a K t t + Y Y | < Y + ( d + r ) ( a z + Y E Z P £ + Y M Z P m + a z t t + Y Y Z Y ) + u) - K_x) + dK, where, 1.3.40) J = Y K K + Y Z Z ( d 2 + r d ) + Y K Z ( 2d+r ) + a d K K ( d 2 + r d ) . Thus I obtain a 6-equation system of equations completely descr ib ing the dynamic behavior of the monopoly f i r m , cons i s t i ng of the va r i ab le fac tor demand equations (1 .3 .30 ) , (1 .3 .31) , (1 .3 .32) , the output p r i ce equation (1 .3 .34) , the demand funct ion (1 .3 .36 ) , and the quas i-f ixed f a c to r accumulation equation (1 .3 .39) . In order to f a c i l i t a t e empir ica l implementation I now make the fo l lowing add i t iona l two assumptions: i ) Short run product ion a c t i v i t y i s cond i t iona l on the c ap i t a l stock at the beginning of the per iod (K_ i ) . Thus, c ap i t a l stock adjustments during the per iod a f f e c t production in the fo l lowing pe r iod , implying that "K" in the equations above must be replaced by " K - l " . i i ) K can be approximated ( " c lose l y enough") by A K = K-K_i, and thus Z can be charac te r ized by AK+dK_1. Est imation of t h i s system of equations resu l t s in parameter estimates f o r the technology which completely speci fy the dynamic demand func t ions ; the very -39-shor t- , short- , and long-run pr i ce e l a s t i c i t i e s can be computed, cons i s ten t with the Marsha l l ian framework. The der i va t ion of these e l a s t i c i t i e s i s s l i g h t l y d i f f e r e n t from that in previous work such as BFW and Morrison-Berndt where output i s assumed g iven , as both subs t i tu t ion and output e f f e c t s here must be taken in to cons ide ra t ion . This under l ies my d i s t i n c t i o n between very-short-run and short-run. I def ine very-short-run as j u s t the d i r e c t subs t i t u t i on e f f e c t with both cap i t a l and output g iven, short-run as the to ta l subs t i tu t i on and output e f f e c t modeled in terms of an output p r i ce change given the demand curve the f i rm faces and given the current cap i t a l s tock, and long-run as the f u l l impact given adjustment to the steady state cap i t a l stock leve l K * . Th is incorporates my d i s t i n c t i o n made above that with the monopoly s p e c i f i c a t i o n there i s an extra degree of freedom for the f irm in the short run as compared to the competi t ive case . Here output p r i ce va r i a t ions can be a "bu f fe r " to exogenous shocks experienced by the f i rm ; the f irm can cushion the otherwise required large changes in input demand by changing i t s output p r i c e . Th is d i s t i n c t i o n can be expressed in terms of se lected representat ive e l a s t i c i t i e s . Here I report the expressions for the very short- , shor t- , and long run energy own pr i ce e l a s t i c i t i e s : 1.3.41 a) e ^ p e = (P E/E) (aE/aP E ) | K = ^ Y = y , b) e | > p e = (P E/E) (3E/aP E j K =K f y=Y + aE/3Y(aY/aP}3P/3P E| K =^) O e E ) P e = (P E/E) (3E/3P E j K = j ^ + 3E/3Y(3Y/3P)3P/3PE + 3E/3K* (3K*/9P E ) • In add i t i on , although the e l a s t i c i t i e s of input demands with respect to output cannot be ca l cu l a t ed with in t h i s model (output i s endogenous), the impact of changes in the demand curve as represented by changes in the exogenous va r i ab l e s , such as income, or a pure s h i f t in the curve -40-charac te r ized simply by a change in the in te rcept of the curve , can be der i ved . Th is i s s im i l a r to an e l a s t i c i t y o f demand with respect to output in the pe r f e c t l y compet i t ive model, for such an e l a s t i c i t y a lso r e f l e c t s changes in demand. Note that the in te rcept term, a, i s not ac tua l l y an exogenous v a r i ab l e ; i t i s a parameter endogenous to the economy but s t i l l exogenous to the f i rm . One can compute the e f f e c t s of a change in th i s output demand re l a t i onsh ip on the f i r m ' s demand for inputs as a chain de r i v a t i v e . No adjustment of pr i ce and quantity i s allowed in the very short run; hence these e l a s t i c i t i e s are def ined only in the short and long run. For example, to determine the short run and long run e f f e c t s of a change in income (INC) on the demand fo r energy one can simply c a l c u l a t e : 1 . 3 . 4 2 ) I N C = (INC/E) ( 3 E / 3 Y ( 3 Y / 3 I N C ) | K = ^ ) , E E INC = ( I N C / E ) (3E/3Y(3Y/3lNC)jK=£- + 3E/3K(3K/3Y)3Y/3INC). The e f f e c t s o f a change in the in te rcept can be ca l cu l a ted s i m i l a r l y . C l e a r l y these e l a s t i c i t i e s provide useful i ns igh t s in to the e f f e c t s of c y c l i c a l demand f luc tua t ions in the economy on output p r i ce and var iab le input demands. The extent of the change in output pr i ce with respect to a change in the exogenous var iab les can a lso be ca l cu la ted with in t h i s model, r e su l t i ng in an output p r i ce e l a s t i c i t y . S im i l a r l y to the e l a s t i c i t i e s of input demand with respect to changes in output demand def ined above, in t h i s case only short run and long run output p r i ce e l a s t i c i t i e s are v a l i d . In terms o f exogenous changes in the p r i ce of energy, the short and long run output p r i ce e l a s t i c i t i e s can be ca l cu l a t ed as: 1 . 3 . 4 3 ) E P S ) P E = P E / P O P / 3 P E ) | K = K , e P,Pe = P E / P { 3 P / a P E ) + 3 P / 3 K * ( 3 K * / 3 P E ) ) . -41-This i s not a conventional demand e l a s t i c i t y , but i s useful fo r p rov id ing imp l i ca t ions of the importance of output p r i ce v a r i a b i l i t y in cushioning shocks. The usual e l a s t i c i t y of demand for output can a lso be ca l cu l a t ed with in th i s model as (aY/aP)(P/Y). This e l a s t i c i t y of demand for output can be used to determine the extent of monopoly power in the indus t ry , as has been discussed in Appelbaum (1977) in terms of the inverse demand e l a s t i c i t y . A l t e r n a t i v e l y , the inverse demand e l a s t i c i t y can be in te rpre ted as the Lerner index of concent ra t ion , which i s used to der ive impl i ca t ions about the s t ructure of the indus t ry . These e l a s t i c i t i e s and other representat ions of the technology and f i r m ' s behavior w i l l be discussed fur ther in Sect ion V. A few f i na l comments should be made about the "genera l " model to be est imated. As in most empir ica l work, very important assumptions are being made regarding the app l i ca t i on of the model, to the data to al low for data l i m i t a t i o n s . The f i r s t assumption concerns the approximation of a continuous 30 model by a d i s c re te empir ica l s p e c i f i c a t i o n . Because o f lack of data a t the appropriate micro leve l the second i s the aggregation of micro dec is ions over the indus t ry ; t h i s impl ies that f irms can be aggregated to an industry l e v e l . Aggregation across time i s an important problem which i s d i f f i c u l t to take into account. The problems involved are both common and d i f f i c u l t . Theoret ica l models are often spec i f i ed in continuous t ime. There are various advantages to t h i s , some of which have been discussed by Koopmans (1950). When dea l ing with an aggregate f i rm in which the components carry out dec is ions with in a d i sc re te time in te rva l which var ies over the u n i t s , the aggregate can be charac te r ized by continuous v a r i a t i o n . However, data ava i l ab le to implement these models emp i r i ca l l y i s necessar i l y d i s c r e t e , often -42-over long i n t e r va l s such as a year . This may create a m i s s p e c i f i c a t i o n •31 problem in terms of adapting the continuous model to d i s c re te t ime. It i s c l e a r that the importance of t h i s type of m i s spec i f i c a t i on w i l l increase as the d i s c re te time i n t e r va l s are l a rge r , so that use of annual data to approximate a continuous r e l a t i onsh ip may reduce r e l i a b i l i t y . The magnitude of the problem cannot be d i r e c t l y i n f e r r e d , and therefore must be 32 assumed to be sma l l . The problem of aggregation across the industry w i l l a lso be set as ide . As in most previous research of t h i s type I wll assume that a micro-based theory o f the f i rm i s an adequate approximation to a theory o f the behavior o f the indus t ry . The only concession fo r t h i s , although i t i s not r igorous ly j u s t i f i a b l e as an aggregation al lowance, i s the incorporat ion of monopol ist ic and some monopsonistic elements. Many important aggregation e f f e c t s are therefore neglected, but l i t t l e work has been done on aggregation in a dynamic model, and what has been done ind ica tes that at t h i s point in time in order to keep the model t rac tab le the s imp l i f y i ng assumptions are necessary to achieve empir ica l a p p l i c a b i l i t y . -43-IV. Empir ical Spec i f i c a t i on and Implementation IV.A. Data For empir ica l implementation of the model, I have used the annual data on pr i ces and quant i t i e s o f Y, K, Z, L, E, and M, the real rate o f r e tu rn , r, and the deprec ia t ion ra te , d, for U.S. manufacturing, 1948-77, out l ined in Appendix C. In a d d i t i o n , the exogenous var iab les X have been spec i f i ed as tota l per cap i ta personal d isposable income in thousands (INC), the unemployment rate of a l l c i v i l i a n employees in percentages (UNEMP), and to ta l populat ion in m i l l i o n s of persons (POP), as reported in the 1980 Economic Report of the Pres ident , tables B-22, B-29, and B-26, r e spec t i v e l y . IV.B. Econometric Considerat ions An important econometric concern with th i s model i s that i t i s a simultaneous system. In add i t i on , on inspect ion i t i s c l e a r that the system i s not r ecu rs i ve , although previous more simple models have been, fo r here Y i s an endogenous va r i ab l e . Thus, est imat ion by simple systems methods w i l l not be cons i s t en t ; the Jacobian term in the l i k e l i h o o d funct ion does not van ish , so that assuming a regular "seemingly unrelated regress ions" l i k e l i h o o d funct ion i s not v a l i d . To see t h i s , wri te the simultaneous equations system in general as : 1.4.1) u y u u u E L P Y-f y (P,X) p- f p ( Y , t ,K ,Z , P . ) (i=E,M) L- f L ( Y , t ,K ,Z , P . ) E - f E ( Y f t , K , Z , P t ) -44-u M = M-f M (Y , t ,K ,Z ,P . ) u z = Z - f z ( Y , t , K , P . , P K , u , d , a ) ) , which in matrix notat ion i s : 1.4.1 ' ) u t = f ( y t , x t , B ) , where represents the endogenous va r i ab l e s , x^ . the exogenous v a r i ab l e s , and B the parameter matr ix . Assume u t i s mu l t i va r i a te normally d i s t r i bu ted with mean vector 0 and constant covariance matrix <j. The general l i k e l i h o o d funct ion fo r t h i s type of problem i s : t=i V V 1.4.2) In L = (-6T/2) ln(2Tr) - (T/2) In |u|+ T In \J | - (1/2) £ where G i s the number of equat ions, T the number of observations in each equat ion, u the variance-covariance matrix of the s t ruc tura l disturbances and |j j the Jacobian which can be expressed as : 1.4.3) J = 3U p /3p 3U L /3p 3U £ /3p auM/ap 3U z /3p 3U y /3p 3Up/3L 3U L /sL 3U E/3L 3U M/3L 3U z /aL 3U y/3L 3Up/aE 3u L /aE 3U E/3E 3UM/3E 3U z /sE 3U y/3E 3U p /3M 3u L/aM 3U £ /3M 3U M/3M 3U Z /3M 3U y /3M 3Up/3Z su^/sZ 3u E /aZ 3u M/aZ 3U Z /3Z 3u y /aZ 3U p /3Y 3U[_/3Y 3U E/3Y 3U M /3Y 3U z/aY 3U y /3Y 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1/B-0 0 0 a Z + Y Z Z z + a Z t t YEZ + Y y Z KZ K, 1 -45-where K p = 6 1 + 2 ( a y + r E y P E + r M y P M ) , K L = V a O t t + V - Y E M P E V • 5 ( Y E E P ^ W M ) + V + Y y Z Z ' K E = a E + Y E M P M + Y E E P E + 2 Y E y Y + a E t > K M = a M + Y E M P E + Y M M P M + 2 Y M y Y + a M t t ' and K z =m* ( ( - l / J ) Y y K +(d+r ) Y y Z ) . Two things should be noted about t h i s Jacobian. F i r s t , i t i s not t r i angu la r and therefore does not vanish from the l i k e l i h o o d func t i on . Hence the s imultanei ty must be taken in to account. A lso s ince these in terdependences are quite complex, t h i s suggests that i t would be very messy to solve fo r the reduced form to condense the system to one which i s amenable to usual maximum l i k e l i h o o d est imat ion procedures. A second impl i ca t ion of t h i s Jacobian i s that i f the endogeneity of Y were not e x p l i c i t l y recognized, so that the f i n a l row and column of the Jacobian could be ignored, the system would remain recu rs i ve . S p e c i f i c a l l y , although Z i s an independent var iab le in a l l the equations and i s in l i nea r form in a l l but the labor demand equation (where i t appears in squared form), the Jacobian i s un i t - t r i angu l a r . Since In | j j appears in the t r ad i t i ona l simultaneous equations l i k e l i h o o d func t ion , in th i s case In j j j =0 at a l l observat ions. Thus i t would not be necessary to take the s imultaneity into account. I t e ra t i ve Ze l lne r est imat ion of the system would be numerical ly equivalent to maximizing the f u l l simultaneous equations l i k e l i h o o d func t ion . S ince in the monopoly model spec i f i ed above Y i s endogenous and since a l l the necessary information to determine both p and Y has been inc luded by incorporat ing the demand equation into the system, a l t e rna t i ve systems est imat ion methods must be used. Three poss ib le so lut ions are immediately 33 ev ident . The f i r s t i s to solve fo r the reduced form to "purge" out the s imultanei ty problem and estimate th i s reduced form system by the usual -46-maximum l i k e l i h o o d methods. This i s , however, as suggested above, very cumbersome a n a l y t i c a l l y and therefore not a very a t t r a c t i v e procedure. The second p o s s i b i l i t y i s to use an instrumental va r iab le technique fo r est imat ing the system, such as Nonl inear Three Stage Least Squares (NL3SLS). Th is procedure in e f f e c t takes the endogeneity of the system into account by c a l c u l a t i n g f i t t e d values of the endogenous var iab les in terms of the exogenous v a r i ab l e s , in order to derive instruments to "p lug i n " to obtain f i n a l parameter est imates. These instruments are not "opt imal " instruments; even i f the system were i t e r a t e d , the instruments are never updated a f te r the f i r s t stage. Thus any over- ident i f y ing cross-equat ion r e s t r i c t i o n s are ignored in formulat ion of the instruments, r e su l t i ng in a numerical incons is tency between the res idual var iance used, depending on the reduced form, and the s t ruc tura l r e s i d u a l s . An addi t iona l disadvantage i s that there i s l i t t l e d i s t r i b u t i o n theory that i s app l i cab le to 3SLS, so that hypothesis t e s t ing i s d i f f i c u l t . Log- l ike l ihood r a t i o s , fo r example, are not v a l i d . It i s poss ib l e , however, although computat ional ly a b i t cumbersome, to c a l cu l a t e Wald s t a t i s t i c s . This i s accomplished by re ta in ing the covariance matrix from the most unconstrained vers ion of a model to use as a bas is fo r imposing cons t r a i n t s . A t h i r d approach i s to use a Fu l l Information Maximum L ike l ihood (FIML) procedure. This takes the s imultanei ty of the system d i r e c t l y in to account by using the system-specif ic Jacobian e x p l i c i t l y to evaluate the re levant l i k e l i h o o d func t i on . This can be in terpre ted as incorporat ing the optimal instruments not ca l cu l a ted in 3SLS to der ive the parameter estimates because i t impl ies that once the f i t t e d values from the reduced form are derived and the parameter estimates based on these r e su l t s are c a l c u l a t e d , the system re-simulates the en t i r e model to obta in new f i t t e d values of endogenous va r i ab les to use as instruments. Thus t h i s procedure in e f f e c t i t e ra tes with -47-respect to both the instruments and the covariance matrix so that in the f i n a l i t e r a t i o n they are cons i s ten t with each other . This procedure i s much more complicated a n a l y t i c a l l y , however, and therefore computational ly much more expensive. Note that the asymptotic proper t ies of these l a t t e r two est imators are equ iva lent , i nd i ca t i ng some ambiguity in choosing between the est imat ion procedures. Although FIML i s t h e o r e t i c a l l y a t t r a c t i v e , NL3SLS has the des i rab le property that i t does not require a normality assumption, which i s e spec i a l l y important for small samples. In add i t i on , 3SLS has an advantage over FIML in the sense that i t can be mechanical ly cons is tent with the ra t iona l expectat ions hypothesis , and therefore the lack of expectat ions s p e c i f i c a t i o n with in the model i s more e a s i l y j u s t i f i e d . S p e c i f i c a l l y , theore t i ca l r e su l t s from Hansen (1981) and Hansen and Singleton (1981), l a t e r discussed and u t i l i z e d in Pindyck and Rotemberg (1982) in a case s im i l a r to the model I have s p e c i f i e d , suggest that the use of 3SLS in e f f e c t "purges" e r rors from the measurement of the expected exogenous var iab les which are co r re l a ted with the e r ror term. This therefore r esu l t s in the er ror-or thogona l i ty property which i s a c ruc i a l component of the ra t iona l expectat ions hypothesis. Although the use of t h i s theore t i ca l equivalence between the s t a t i s t i c a l and theore t i ca l model impl i ca t ions provides a model that w i l l not be biased by m is-spec i f i ca t ion of the non-static expectat ions process, i t does not of course provide an e x p l i c i t model of expectat ions format ion. Th is can be in terpreted by cons ider ing pr i ces in the s t a t i c expectat ions model—which by d e f i n i t i o n are expected to be constant over time so that e r rors occur in current expectat ions with respect to any future exogenous va r i ab l e—to be measured with e r ro r . Thus, in terms of econometrics, there i s -48-an e r ro rs- in-va r i ab les problem to be dea l t with rather than an e x p l i c i t s p e c i f i c a t i o n o f an expectat ions formation process to be cha rac te r i zed . To state th i s more c l e a r l y in terms of the usual model in which only current va r iab les are considered important, I shown in Essay 2 that non-stat ic expectat ions about future exogenous va r i ab les can be in te rpre ted as a " d e f l a t i o n " of the current va r iab le used for decision-making purposes. I.e., i f the pr i ce of energy were expected to increase r e l a t i v e to other p r i c e s , current dec is ions about the future path of stock var iab les must be made with th i s knowledge taken in to account. One way the f i rm can accomplish t h i s i s by making current dec is ions based on the cur rent p r i ce plus some weighted average of future expected pr i ces which " de f l a t e s " the current value to take these expectat ions of future changes in to account. I f t h i s " d e f l a t i o n " were not taken in to account, there would be an e r ro r in the s p e c i f i c a t i o n or measurement o f the re levant exogenous va r i ab l e . A model depending on var iab les measured with e r ro r cannot be estimated i n the same way as i f they were not, for the measured exogenous var iab les are not d i s t r i bu t ed independently of the s tochast i c disturbance term. S p e c i f i c a l l y , the disturbance term w i l l inc lude components depending on the measurement e r ro r and the c o e f f i c i e n t s o f the var iab les measured with e r r o r . Least squares estimates in t h i s case w i l l not only be biased but i n cons i s t en t , suggesting that the i n t e rp re ta t i on of non-stat ic expectat ions as an e r rors in var iab les problem i s ser ious not only i n t u i t i v e l y and t h e o r e t i c a l l y , but a lso s t a t i s t i c a l l y . The best way to adjust for t h i s type of problem would obviously be to use " co r r e c t " measures of the exogenous va r i ab l e s , i . e . , va r iab les adjusted co r r e c t l y fo r the expectat ions " d e f l a t i o n " . This i s the bas is of Essay 2. A second-best so lu t ion i s to use an instrumental va r iab les approach to the er rors in va r iab les problem. -49-The c r i t e r i o n for instrumental va r i ab les i s that they should not have any c o r r e l a t i o n with the e r ro r term but a high c o r r e l a t i o n with the var iab les that they are to rep lace . Th is can be accomplished by choosing exogenous var iab les excluded from the system, or the equation i f s ing le equations are being cons idered. Two such a l t e rna t i ves were used by Pindyck and Rotemberg (1982). A l l o f t he i r equat ions—a system of a cos t funct ion and share equations fo r a fac tor demand model—include the current values of the exogenous va r i ab l e s . Thus, other va r i ab les outs ide of the model, or lagged values of the exogenous var iab les were used as a l t e rna t i ve " cond i t i on ing s e t s " , or groups o f instrumental v a r i ab l e s . Pindyck and Rotemberg show, from Hansen and S ing le ton , that est imat ion using these instrumental va r iab les i s t h e o r e t i c a l l y cons is tent with ra t iona l expectat ions , and that the r e su l t i ng "genera l ized method o f moments" est imator i s equiva lent to three stage l eas t squares with these var iab les as instruments. Note that t h i s procedure i s conceptual ly equiva lent to the usual est imat ion approaches used fo r simultaneous equat ions. In the e r rors in var iab les case enough excluded exogenous var iab les must be used to a l low " i d e n t i f i c a t i o n " of the e r ro r to "purge" i t from the re levant v a r i ab l e s . Th is i s s im i l a r to the simultaneous equations case , in which enough exogenous var iab les in the system must be excluded from each equation to s a t i s f y the rank and order cond i t ions fo r i d e n t i f i a b i 1 i t y . 3SLS a lso provides cons is tent estimates in a simultaneous equations system using a l l of the exogenous var iab les as instruments as long as enough exogenous var iab les are inc luded in the system but excluded from each equation to i d en t i f y the parameters. In t h i s sense, the est imat ion procedure i s s im i l a r to that suggested by Pindyck and Rotemberg, and thus i s a lso cons i s ten t with ra t iona l expectat ions as i t in e f f e c t "purges" the equations -50-o f the e r ro r that i s co r re l a ted with the disturbance term, and which i s in terpre ted as the i m p l i c i t impact of non-stat ic expectat ions . Note that the choice of instrumental var iab les used for the ra t iona l expectat ions j u s t i f i c a t i o n for 3SLS in which the exogenous var iab les themselves are e x p l i c i t l y " f i t t e d " with the instrumental var iab les procedures, may be d i f f e r e n t . S p e c i f i c a l l y , the exogenous var iab les subject to expectat ions formation--the r e l a t i v e input pr ices—would not be used as instrumental var iab les in the expectat ions r a t i o n a l i z a t i o n . However, output i s endogenous. Th is i s a lso i n t u i t i v e l y the most important to model for a non-stat ic expectat ions model, as the input pr i ces are in r e l a t i v e terms rather than abso lute . Thus the 3SLS j u s t i f i c a t i o n s t i l l ho lds . In add i t i on , in order to i d e n t i f y the simultaneous equations model enough excluded exogenous var iab les are necessary in the demand equation to i den t i f y the res t of the equat ions. This in e f f e c t inc ludes these exogenous var iab les in the system but not i n the var iab le input demand equations or cap i t a l accumulation equat ions, s i m i l a r l y to the case of Pindyck and Rotemberg. I be l ieve that t h i s argument provides a strong case fo r using 3SLS for est imat ion instead of FIML. In add i t i on , p a r t i c u l a r l y with the external cos ts o f adjustment model, t h i s model may be too " p a r t i a l " to be v a l i d l y estimated by a procedure that assumes a l l information to be ava i l ab le in the system. Other important r e l a t i o n s h i p s , such as the behavior of the cap i t a l goodswmarket which determines the shape o f the external costs func t i on , are not modeled. I f "outs ide" va r iab les were necessary for the expectat ions formation and input dec i s ion processes, the model would c l e a r l y not be f u l l y s p e c i f i e d . Thus I have used 3SLS fo r est imat ion i n t h i s essay. Note that a l l o f the preceding and fo l lowing d iscuss ion of est imat ion depends on the i d e n t i f i c a t i o n of the parameters of the simultaneous system. This can be checked in an equation-by-equation "count ing" sense by assess ing the usual rank cond i t i ons . -51-Given that the only endogenous van 'b les enter ing the r i gh t hand s ide o f the equations are Z and Y, the maximum number of endogenous var iab les in any s ing le equation i s three . Thus to ensure that in every equation the number of excluded exogenous va r i ab les i s at l e as t as large as the number o f inc luded endogenous v a r i ab l e s , i t i s necessary to have three excluded exogenous v a r i a b l e s . 3 ^ The parameters o f the system are thus estimated by 3SLS under the assumption that a disturbance term could simply be added to each equat ion, r e su l t i ng in a disturbance vector that i s independently and i d e n t i c a l l y mul t ivar ia te-normal ly d i s t r i bu t ed with mean vector zero and constant non-singular covariance matrix u. V. Empir ical Results -52-V.A. Est imat ion and Parameter Estimates As discussed at some length in the previous sec t i on , est imat ion o f t h i s model was c a r r i ed out using i t e r a t i v e non-l inear three stage l eas t squares (NL3SLS) on the system of equations represented by (1 .3 .30) , (1 .3 .31 ) , (1 .3 .32 ) , (1 .3 .34 ) , (1 .3 .36 ) , and (1.3.40) in Sect ion III. Due to a l im i t ed amount of data , in some cases convergence was d i f f i c u l t to achieve and i t was necessary to impose cons t ra in ts on ce r t a in s t a t i s t i c a l l y i n s i g n i f i c a n t parameters to obtain f i n a l est imat ion r e s u l t s . In p a r t i c u l a r , i t was necessary in general to const ra in Y k z and Y y z t 0 ze ro , both of which appeared i n s i g n i f i c a n t l y d i f f e r e n t from zero in unres t r i c t ed runs. In add i t i on , in some 1949-77 spec i f i c a t i ons Y ^ became i n s i g n i f i c a n t l y d i f f e r e n t from zero and appeared v o l a t i l e . Therefore was constra ined to the value which appeared quite robust within the 1948-71 s p e c i f i c a t i o n s , . 05 . I now pursue d iscuss ion of empir ica l r e su l t s by examining each of the extensions of t h i s model in tu rn . For each subsection I proceed by f i r s t cons ider ing the most general s p e c i f i c a t i o n of the model, the gross investment model with both in terna l and external costs of adjustment of the 4-input (K,L,E,M) formulat ion with one quas i- f ixed f a c to r , K, over the en t i r e time per iod 1948-77. I then compare these r esu l t s to those obtained using more r e s t r i c t i v e spec i f i c a t i ons on costs of adjustment, only in terna l or external costs of adjustment and net rather than gross investment, as these spec ia l cases have been common in recent l i t e r a t u r e . I p a r t i c u l a r l y emphasize the model of in terna l costs of adjustment on gross investment, for t h i s i s my "p re fe r red " s p e c i f i c a t i o n . F i n a l l y , I b r i e f l y compare my r esu l t s for 1948-77 to analogous estimates fo r se lected models over the time per iod o f the -53-TABLE 1-1 Parameter Estimates with A l t e rna t i ve Spec i f i c a t i ons on Investment and Adjustment Costs , 1948-77 (Asymptotic t - s t a t i s t i c s i n parentheses) Gross Investment Internal and Gross Investment Gross Investment Net Investment External Costs Internal Costs External Costs Internal Costs 6 ay a 0 a 0 t aE a M YEM YEY a E t aMt YYK YEE .0009 .0010 .0006 .0009 (10.01) ( 8.81) ( 9.44) (10.74) -.0010 .000003 -.000005 -.0003 (-5.68) ( .02) ( -.05) (-3.95) 1.389 .885 .745 .752 (16.96) (11.11) (16.94) (18.39) .044 .026 .012 -.0018 (10.23) (7.48) (4.44) (-.80) .042 .076 .049 .041 (5.52) (6.62) (7.41) (6.55) .755 1.037 .904 .756 (16.02) (22.23) (23.76) (19.46) .0023 .016 .0023 .0025 ( .56) (3.14) ( .54) ( .60) -.00003 -.0001 -.00005 -.00004 (-5.50) (-11.46) (-7.25) (-6.86) -.0004 -.0006 -.0004 -.0003 (-7.13) (-8.74) (-8.05) (-5.88) .0006 .0022 .0008 .0008 (4.37) (11.11) (5.33) (5.19) .0078 .010 .006 .0043 (6.40) (6.46) (5.37) (4.02) -.0055 -.009 -.0049 -.0019 (-9.50) (-11.95) (-8.04) (-4.05) -.0004 -.005 -.0021 -.0020 (-.26) (-2.47) (-1.39) (-1.32) .019 -.008 .022 .0098 (1.23) (-.40) (1.44) ( .56) -54-Y Z Z a K a K t «Z a Z t YEK Y EZ YMZ dKK TABLE I-l (cont 'd) Parameter Estimates fo r A l t e rna t i ve Spec i f i c a t i ons on Investment and Adjustment Costs , 1948-77 Gross Investment Internal and Gross Investment Gross Investment Net Investment External Costs Internal Costs External Costs Internal Costs 62,POP 62.UNEMP .05 .05 .031 .0080 (8.90) (3.16) .599 .504 1.422 (3.98) (3.92) (12.04) -1.800 -1.319 -.850 -.925 (-7.58) (-3.55) (5.72) (-5.64) -.139 -.056 -.058 .026 (-8.76) (-2.80) (-6.28) (2.64) -10.181 -1.477 (-4.84) ( -.55) -.217 -.366 (-2.17) (-3.16) .036 -.090 .017 .035 (2.59) (-4.45) (1.31) (2.91) -.151 -.023 (-3.03) (-.22) -.286 -1.037 -.795 -.395 (-2.42) (-7.21) (-8.76) (-3.70) -.669 .022 (-1.68) ( .03) -.046 .136 (-.66) (5.14) 2.800 3.268 3.181 3.153 (18.50) (32.36) (29.47) (29.90) -.0077 -.014 -.012 -.011 (-5.32) (-14.52) (-11.91) (-10.47 .019 .008 .017 .012 (3.36) (1.89) (3.52) (2.57) -55-TABLE I-l (cont 'd) Parameter Estimates fo r A l t e rna t i ve Spec i f i c a t i ons on Investment and Adjustment Cos ts , 1948-77 Gross Investment Internal and Gross Investment Gross Investment Net Investment External Costs Internal Costs External Costs Internal Costs 62,INC .119 .367 .206 .184 (3.23) (8.62) ( .47) (6.16) R?Js P .8465 .9164 .9692 .9851 L .8363 .8588 .9323 .8021 E .9958 .9490 .9879 .9913 M .9914 .9801 .9854 .9924 Z .8410 .9125 .8080 .0123 Y .8950 .9617 .9074 .9160 |x \{1971) .304 .317 .470 .052 -56-on 'g ina l Berndt-Wood study, and the fo l lowing o r ig ina l dynamic studies of Berndt Fuss and Waverman (1979) and Pindyck and Rotemberg (1982), 1948-71, as th i s foundation in the l i t e r a t u r e makes i t a useful bas is for contrast and comparison. Before d i scuss ing the information ava i l ab le from th i s extended model, I be l ieve i t i s useful to consider parameter estimates from a l t e rna t i ve formulat ions. These are presented in Tables 1-1 for the 1948-77 estimates and 2 1-2 fo r analogous se lected 1948-71 s p e c i f i c a t i o n s . Note f i r s t that R s are 2 reported for the ind i v idua l equations even though R s are not well def ined i n the case of 3SLS. This problem ar i ses because the " f i t t e d " equations are formed us ing the f i r s t stage " f i t t e d " values o f endogenous va r i ab l e s , which therefore incorporate the corresponding disturbances instead of the actual data. 2 2 The reported R ' s , therefore are "pseudo-R 1 s " ; they simply represent 2 the R of an ordinary least-squares regress ion of the actual on the f i t t e d 2 va lues , the simple co re l a t i on c o e f f i c i e n t squared. The r e su l t i ng R 's are in general qui te h igh. It i s d i f f i c u l t to choose a "p re fe r red " s p e c i f i c a t i o n on the bas is of these est imates, as which s p e c i f i c a t i o n i s "be t te r " depends on the equation chosen for emphasis. It i s c l e a r , however, that investment, which i s genera l ly d i f f i c u l t to exp l a i n , i s best charac te r ized by the model with interna l costs on gross investment (the gross-internal model), whose estimates appears in the second column of Table 1-1. Focussing on t h i s s p e c i f i c a t i o n , as i t w i l l become c l ea r l a t e r that t h i s appears to be the most representat ive formula t ion , one can cons ider a l t e rna t i ve i nd i ca t i ons of the model 's behavior . For example, the " f i t t e d values" for f l uc tua t ions in both K and E, which have been notor ious ly d i f f i c u l t to capture for the 1970's , appear to bet ter represent actual demand than most previous models. S p e c i f i c a l l y , f ac tor demand models in general have -57-not r e f l e c t ed the substant ia l dec l ine in investment in the " cap i t a l formation slowdown" o f 1974, and have tended to over-predict the pr i ce responsiveness of energy between 1973 and 74, exaggerating the f a l l in energy demand from the 1973 energy pr i ce shocks. This model fares qui te well in model l ing t h i s behavior. F i t t e d values of the E and Z equations for 1970-77 are reported below: Table I . l .A . True and F i t t e d Values of E and : Year E E Z 1 1970 30.60 28.58 21.59 19.53 1971 29.49 28.69 22.84 22.04 1972 31.06 29.56 26.81 26.12 1973 31.93 29.81 28.77 28.58 1974 33.12 31.77 23.48 23.49 1975 31.31 33.22 28.04 26.69 1976 33.66 33.03 30.62 28.18 1977 34.35 32.88 31.76 29.11 Z, gross-internal model, 1970-77. The r e l a t i v e values actual (Z) and f i t t e d (Z) ind i ca te that the 1974 slowdown in investment i s captured well wi th in th i s model, contrary to most previous models. Th is helps to j u s t i f y using th i s s p e c i f i c a t i o n for model l ing the determinants of investment, and therefore as a basis for ana lys i s of investment i nd i ca to r s such as Tob in ' s q, or phenomena such as the p roduc t i v i t y slowdown of the 1970's . The behavior of energy demand i s a lso well charac te r ized by t h i s model, p a r t i c u l a r l y the increase rather than decrease in energy demand in 1974 over 1973, contrary to what most models have predic ted based on the large energy p r i ce increases in 1973. S p e c i f i c a l l y , the actual increase in energy demand between 1973 and 1974 was 1.19 as compared to an estimated change of 1.96, or in percentage terms approximately 3.6 percent and 6.2 percent r e spec t i v e l y , In con t ras t , BFW report a predicted decrease in energy demand of 2 percent using the i r dynamic model and 10.3 to 19.6 percent using a l t e rna t i ve s t a t i c equ i l ib r ium models. -58-The ind i ca to r s discussed here suggest that the monopoly model s p e c i f i c a t i o n , p a r t i c u l a r l y that based on the gross investment model with in terna l costs of adjustment, has enough f l e x i b i l i t y to capture the fac tor demand patterns of the v o l a t i l e 1970's rather we l l . Problems involved in modeling the post-1971 time per iod have caused some researchers to assume that conventional econometric models are of l i t t l e use to analyze post-1973 developments. Others such as Pindyck and Rotemberg (1982) simply neglect post-1973 data and use the e a r l i e r data since i t i s stable enough to e a s i l y provide p l aus ib l e r e s u l t s . The resu l t s here for the 1948-77 time per iod therefore appear to provide substant ia l support for t h i s type of model. The parameter estimates reported in Table I-l can be fur ther analyzed to provide more s p e c i f i c information on the a p p l i c a b i l i t y of t h i s model to the data . The f i r s t po int to consider i s that many of the signs of these parameters are r e s t r i c t e d by economic theory. From BFW (1979) and Diewert (1978) the main r e s t r i c t i o n s impl ied by economic theory can be stated as : Y E E ' YMM < ° » YKK* Y ZZ > °> B l > 0 ' These r e s t r i c t i o n s are necessary to obtain negative own pr i ce e l a s t i c i t i e s for the var iab le inputs , a pos i t i v e adjustment c o e f f i c i e n t x , and a downward s loping demand curve fo r output. In add i t i on , BFW (1979) asser t that y Y K must be negat ive . In t he i r work output i s exogenous so th i s i s equiva lent to assuming a pos i t i v e cap i ta l-output e l a s t i c i t y . In t h i s model e l a s t i c i t i e s with respect to output are not de f ined , as output i s endogenous. Instead t h i s impl ies in terms of the technology that aK*/3Y>0, or that the cap i t a l input i s a "normal" input in production rather than i n f e r i o r . As th i s seems i n t u i t i v e l y reasonable, t h i s r e s t r i c t i o n a lso should ho ld . From Tables I-l and 1-2, note that the r e s t r i c t i o n on Y e e holds everywhere, although the r e s t r i c t i o n on y^ holds only in the gross-internal s p e c i f i c a t i o n fo r the 1948-77 time per iod . The remaining models exh ib i t a pos i t i ve y M M , -59-suggest ing that the pure subs t i tu t ion e f f e c t for mater ia ls i s pos i t i v e—not an i n t u i t i v e l y p l aus ib l e assumption. Note, however, that in no case i s t h i s v i o l a t i o n " s i g n i f i c a n t " in the sense that the c o e f f i c i e n t i s never s t a t i s t i c a l l y s i g n i f i c a n t l y d i f f e r e n t from ze ro . The remaining r e s t r i c t i o n s hold throughout a l l s p e c i f i c a t i o n s . F i n a l l y , note that an overview of the 1948-77 as contrasted with the 1948-71 estimates (see Table 1-2) ind ica tes that although the magnitudes of some parameters change s l i g h t l y , in general the signs of the c o e f f i c i e n t s are s im i l a r for both time per iods . As th i s was genera l ly true across various investment and costs of adjustment s p e c i f i c a t i o n s , only the "most" and " l e a s t " general spec i f i c a t i ons are presented for 1948-71, and fur ther r e su l t s based on these models w i l l not be emphasized. Further ana lys i s o f the signs and magnitudes of these r e s t r i c t e d parameters as well as others w i l l be c a r r i ed out in the fo l lowing sec t i ons . Given the cons iderat ions on the parameter estimates already discussed in th i s subsect ion , however, the monopoly model and p a r t i c u l a r l y the gross- interna l model, appears to accord well both with basic economic theory and with actual observed phenomena. V.B. Costs of Adjustment Add i t iona l parameter estimates can be examined to provide evidence on the existence of costs of adjustment. A s i g n i f i c a n t advantage of the most general s p e c i f i c a t i o n , that of gross investment with interna l and external c o s t s , i s that i t permits r igorous t e s t i ng of assumptions made by other researchers in recent work. Berndt-Fuss-Waverman (1979) and Morrison-Berndt (1981), fo r example, allowed in te rna l costs of adjustment on net investment, and assumed that external costs of adjustment were zero . By cont ras t , Pindyck and -60-B «Y a 0 a 0 t a E a M Y£Y YMY a E t a M t Y K YEE YMM TABLE 1-2 Parameter Estimates for Selected Spec i f i c a t i ons on Investment and Adjustment Costs , 1948-71 Gross Investment Internal and Net Investment External Costs Internal Costs .0017 .0022 (9.33) (11.20) -.00094 -.0007 (-5.05) (-3.74) 1.402 .897 (11.82) (8.34) .033 .042 (7.42) (10.34) .074 .060 (7.69) (5.64) 1.056 1.486 (13.84) (18.87) .0093 .0150 (1.82) (2.53) -.00011 -.0001 (-13.40) (-13.60) -.00053 -.0007 (-6.89) (-8.60) .0016 .0023 (5.85) (7.56) .0018 .0014 ( .81) ( .55) -.0074 -.0099 (-10.11) (-8.97) -.018 -.0092 (-5.02) (-2.76) -.208 -.318 (-4.60) (-7.41) -61-TABLE 1-2 (cont 'd) Parameter Estimates fo r Selected Spec i f i c a t i ons on Investment and Adjustment Costs 1948-71 Gross Investment 1 Internal and Net Investment External Costs Internal Costs a K a K t a Z a Z t Y EK ^EZ YMZ dKK B2.P0P B2,UNEMP .05 .053 (8.40) 1.289 1.004 (4.76) (7.04) -1.908 -2.365 (-4.90) (-9.14) -.044 -.0295 - (-1.94) (-1.77) -19.311 (-5.38) -.397 (-2.28) .041 -.0026 (2.16) (-.20) .051 (.68) -.028 -.567 (-3.08) (-3.80) 1.012 (2.07) -.418 (-3.08) 2.938 2.610 (28.54) (23.99) -.0082 -.0042 (-7.23) (-3.91) -.0289 -.038 (-4.55) (7.00) -62-TABLE 1-2 (cont 'd) Parameter Estimates fo r Selected Spec i f i c a t i ons on Investment and Adjustment Costs 1948-71 Gross Investment Internal and Net Investment External Costs Internal Costs 6 2,INC .309 .271 (5.46) (5.45) RfiVs P .9018 .8838 L .7534 .6611 E .9328 .9493 M .9821 .9671 Z .7153 .2969 Y .9852 .9886 A | (1971) .249 .203 - 6 3 -iinvestment and assumed that in terna l costs were zero . Within the general model these spec ia l cases can be tested simply by parameter r e s t r i c t i o n s . For the BFW and M-B s p e c i f i c a t i o n s the re levant hypotheses would be: V d K K = 0 ' H l V 0 ' while fo r PR they would be: H 0 : Y E Z = Y M Z = a Z = a Z t = Y Z Z = 0 ' H l : ^ Z ^ M Z ^ Z ^ Z t ' ^ 0 ' As seen in the f i r s t column of Table 1-1, the BFW and MB nul l hypothesis 37 i s not r e j e c t ed ; the t - s t a t i c t i c i s - . 6 6 0 . By con t ras t , in terna l costs of adjustment are 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 t - s t a t i s t i c s on ( 3 . 0 2 6 ) , Y M Z ( - 1 . 6 8 3 ) , a z ( - 4 . 8 4 2 ) , a Z t (-2.167) , and Y Z Z (3.981) a re , with only one except ion, s t a t i s t i c a l l y s i g n i f i c a n t , suggesting that a j o i n t t e s t of 38 the nu l l hypothesis would lead to r e j e c t i o n . Thus i t appears from t h i s empir ica l ana lys i s that in te rna l cos ts of adjustment have a f i rmer empir ical bas is than do external costs of adjustment. The parameters in Table 1-1 a lso provide information concerning the shape of the adjustment cost func t ions . For adjustment to be gradual rather than instantaneous, i t was argued in the f i r s t sect ion that tota l adjustment costs must be convex in investment. This i s c l e a r l y s a t i s f i e d in a model with both in terna l and external costs i f both funct ions are convex, although that would not be necessary. In add i t i on , i f each of the in terna l and external adjustment costs assumptions were used i nd i v i dua l l y to r a t i o n a l i z e non-instantaneous adjustment, each funct ion would need to be convex. Rothschi ld (1971) has noted that decreasing costs are j u s t as p l aus ib l e a p r i o r i as are increas ing adjustment c o s t s , and that u l t imate ly t h i s issue on the shape of the adjustment cos t funct ions i s an empir ica l one. Using the general model I can exp lo i t the f l e x i b i l i t y of the adjustment cos t -64-s p e c i f i c a t i o n to examine the question of whether tota l costs and each of i t s two components are convex, simply by looking at the signs of the re levant parameters. For external costs to be convex, given the quadrat ic approximation to the external costs funct ion i t i s necessary and s u f f i c i e n t that d K K > 0 . As seen in the f i r s t column o f Table I-l, however, the estimated d ^ i s - .046, suggesting concave rather than convex external adjustment cos t s . This impl ies that external costs alone are not a good j u s t i f i c a t i o n for slow investment behavior . Note, however, that a confidence in te rva l for d ^ would inc lude 3 9 pos i t i v e va lues . In terms of in terna l cos t s , the pos i t i ve c o e f f i c i e n t on Y Z Z » where 2 2 Y Z Z = 3 1 > s a ' ' S 0 s i 9 m ' f i c a n ' t ( a t - s t a t i s t i c of 3.981) implying s t a t i s t i c a l l y s i g n i f i c a n t convexity of in terna l adjustment cos t s . The imp l i ca t ions o f the estimates o f the parameters determining the shape of the adjustment cost funct ion can be fur ther explored in terms of t he i r economic i n t e rp re ta t i on and j u s t i f i c a t i o n . Conventional arguments about the poss ib le convexity of external costs of adjustment are based on the idea that a premium must be paid by a f i rm that attempts to increase the demand for i t s c a p i t a l , as the supply curve fo r a cap i t a l producing industry should be upward s l op ing . Furthermore, the convexity argument says that t h i s premium i s not only p o s i t i v e , i t increases with the s i ze o f the increase in demand, which i s necessary fo r i t to be a 40 cause of slow adjustment behavior for c a p i t a l . This second a s s e r t i o n , which i s the basis of the convexity argument and which i s represented by the d ^ term , i s c e r t a i n l y open to quest ion , and even the f i r s t asser t ion may be i n v a l i d . In terms of the convexity argument, there may not be increas ing costs assoc iated with purchasing cap i t a l on a large s ca l e , p a r t i c u l a r l y over t ime, -65-i f : ( i ) the cap i t a l industry i s charac te r ized by rap id l y expanding technica l change, ( i i ) the cap i t a l industry technology can be represented by a funct ion with s i g n i f i c a n t returns to sca le 4 ' 1 ' , and/or ( i i i ) , i f these or f i n anc i a l or other reasons d i c t a t e there are economies assoc iated with buying c a p i t a l on a large s ca l e , Th is may, fo r example, r e s u l t i f c ap i t a l i s spec i a l i z ed so that producing one un i t to spec i f i c a t i ons i s expensive, but add i t iona l equiva lent uni ts can then be added cheaply . Th is i n turn may be caused by the necessi ty of having to develop ce r t a i n equipment, technology or labor s k i l l to produce the f i r s t un i t of spec ia l order equipment, whereas th i s technology or s k i l l i s therefore immediately ava i l ab le fo r subsequent u n i t s . The U.S. cap i t a l producing industry i s l i k e l y to be charac te r ized by these t r a i t s , which are r e f l e c t e d , e . g . , in the volume discounts ava i l ab le fo r many products . These arguments can thus be used not only to j u s t i f y a concave external costs of adjustment funct ion but perhaps even a downward s loping func t i on . An addi t iona l argument can be made which i s based on the concept of the external costs funct ion as a representat ion of monopsony. In the pe r fec t l y compet i t ive input market case , the c ap i t a l supply funct ion would be pe r fec t l y e l a s t i c — n e i t h e r upward nor downward s l op ing . I n t u i t i v e l y , using est imat ion r esu l t s based on an indust ry , monopsonistic impacts would l i k e l y have an e f f e c t . However, these forces may r e su l t in e i ther an upward or downward s lop ing func t ion . F i n a l l y , i t i s worth emphasizing that the s t a t i s t i c a l i n s i gn i f i c ance of the d K K parameter ind ica tes ambiguity about the importance of the external cos ts o f adjustment parameter. Th is i nd i ca t i on o f the s t a t i s t i c a l l y inconsequential impact of external c o s t s , coupled with the apparent concavi ty o f the external cos t funct ion and the small magnitude of the c o e f f i c i e n t , therefore suggests that costs of adjustment models based s t r i c t l y on external costs of adjustment should be c a l l e d in to quest ion. -66-By con t ras t , the s i gn i f i c ance and s i ze of the in te rna l cos ts parameters, along with the " co r rec t " s ign on Y z z f o r convexi ty , ind ica te that in terna l costs may be more important than external costs in the gross investment s p e c i f i c a t i o n as an explanat ion of slow adjustment of the c ap i t a l stock. Th is can a lso be i n t u i t i v e l y j u s t i f i e d in terms of the increas ing complexity involved as technica l change expands in the cap i t a l indust ry . If c ap i t a l equipment were becoming inc reas ing ly technica l and la rger s c a l e , r equ i r ing more scarce s k i l l e d labor f o r the machinery's i n s t a l l a t i o n , upkeep, and use, t h i s could imply increas ing costs fo r large increments of c a p i t a l . Although in terna l cos ts and not external costs appear to exh ib i t the proper t ies cons i s ten t with slow adjustment, note that the shape of to ta l adjustment costs has not y e t been determined. If the convexity of in te rna l cos ts outweighs the concavity of external costs requirements fo r the convexity o f to ta l adjustment costs are emp i r i ca l l y supported. An a l t e rna t i ve p ropos t i i on , a lso supported by these r e s u l t s , i s that costs may be f i r s t concave or l i n e a r and then become convex before the des i red investment leve l i s reached. In t h i s case , adjustment to the point where convexity appeared would be immediate, but once increas ing costs set i n , adjustment would once again be slowed. Note that the adjustment c o e f f i c i e n t x i n t h i s case i s estimated to be approximately .30, which al lows fo r the f i r s t increments of investment to d isp lay any kind of adjustment costs as long as marginal cos t begins to increase before 30 percent of the "des i r ed " investment i s reached. An a l t e rna t i ve i n t e rp re ta t i on of t h i s i s that investment costs must be convex at investment l eve l s higher than the ex i s t i ng l e v e l , so that the important cons idera t ion appears to be that the quadratic costs assumption i s l o c a l l y v a l i d . F i n a l l y , note that t h i s en t i r e d iscuss ion has been framed in the context of costs of adjustment on gross rather than net investment. S imi la r -67-spec i f i c a t i ons were der ived and estimated fo r net investment, but y i e l d e d quite negative r e s u l t s . Spec i f i c a t i ons with both external and in terna l cos t s , and only external c o s t s , could not be estimated e a s i l y as the term within the square root in the adjustment c o e f f i c i e n t x became negat ive, terminat ing the est imat ion procedure. Th is ind ica tes a s t rongly negative d ^ term. For the few s p e c i f i c a t i o n s for which I was able to obtain est imates, the r e su l t s were d i sappo in t ing . This i s cons is tent with i n t u i t i o n , however, as i t would seem that in terna l costs would be more important than external for net investment. Even the in terna l costs s p e c i f i c a t i o n on net investment, although i t provided the bas is for many previous studies inc lud ing BFW and MB, d id not appear to be as a t t r a c t i v e a s p e c i f i c a t i o n o f f i rm behavior as d id the gross investment models. S p e c i f i c a l l y , as can be seen in Table 1-1, the K equation has an 2 extremely low R of .0123 and the x c o e f f i c i e n t i s a lso very low at .052. 2 This i s in cont ras t with the high R 's on K and p laus ib le adjustment c o e f f i c i e n t s of approximately 30 percent for the gross-internal and gross-internal and external s p e c i f i c a t i o n s , although the gross-external s p e c i f i c a t i o n tends to e r r in the opposite d i r e c t i on with a x of almost 50 2 percent and a r e l a t i v e l y low R on the K equat ion. Thus I w i l l concentrate p r imar i l y on the gross investment s p e c i f i c a t i o n s for the res t of the 4 2 d iscuss ion on r e s u l t s . Above I h igh l ighted the importance of the shape of tota l adjustment c o s t s . The appearance and comparison of these adjustment cos t funct ions can be obtained by graphing the func t ions . The interna l cost of adjustment funct ion i s der ived simply as the sum of components of the var iab le cos t funct ion that depend on investment. The external cos ts funct ion i s represented as a quadrat ic funct ion given by (1.3.37) in Sect ion III. These funct ions have been graphed fo r var ious model spec i f i c a t i ons from the vantage point of the 1977 leve l of investment. The resu l t s in the graphs in F igures -68-Figure 1-1 1J 87.95 T / 869.051 550.164 231.26 + -8T-.63-30.00 38.00 46.00 54.00 > 62.00 INTERNAL AND EXTERNAL: C0STS 0F ADJUSTMENT 0N GR0SS INVESTMENT -69-Figure 1-2 11^0.32 T 820.27J 540.21 + 260.16 + / / / / / / / / / / / / / / / / / / -19.90 30.00 38.00 46.00 54.00 T0TAL C0STS 0F:ADJUSTMENT 0N GR0SS INVESTMENT 62.00 -70-Figure 1-3 11S§.67-914.36 + 630.051 345.74+ / / 61143 30.00 38.00 46.00 : 54.00 62 INTERNAL C 0 S T S 0F ADJUSTMENT, INTERNAL C0STS 0NLY. M0DEL -71-Figure 1-4 2(9.17 j x • 209.554 159.934 110.30 4 60.68 / / / / / / / / / / / / 30.00 : 38.00 46.00 54.00 62.00 EXTERNAL C0STS 0F ADJUSTMENT, EXTERNAL C0STS 0NLY M0DEL -72-Figure 1-5 14*5.89r 1091.92-727.95+ 363.97+ 0.00 0.00 8.00 16.00 24.00 INTERNAL C 0 S T S 0F ADJUSTMENT,. NET INVESTMENT M0DEL \ 32.00 -73-1-5 show, using the 1977 values of the exogenous va r i ab l es , what would occur i f investment were to increase above the e x i s t i n g l e v e l . F igure 1 represents in terna l and external costs of adjustment i n d i v i d u a l l y for the s p e c i f i c a t i o n incorporat ing both types of costs of adjustment on gross investment, and Figure 2 shows t he i r to ta l impact. F igures 3-5 show the form of i n t e r n a l , ex t e rna l , and in terna l costs ( tota l costs ) fo r the g ross- in te rna l , gross-externa l , and net-internal spec i f i c a t i ons r espec t i ve l y , for purposes of comparison and cont ras t . Consider f i r s t the in terna l and external cos t funct ions in F igure 1. It i s c l e a r that i f investment were at a higher level the f i rm would incur pos i t i v e and inc reas ing in te rna l costs o f adjustment but negative and decreasing external c o s t s . The e f f e c t of the in terna l costs overr ides that o f external cos t s , however, r e su l t i ng in the pos i t i v e and increas ing to ta l 43 adjustment costs p i c tu red in F igure 2. The f igures therefore imply that in terna l costs o f adjustment on gross investment have the cor rec t proper t ies to cause slow and caut ious investment by f i rms , whereas external costs not only do not possess these p rope r t i e s , they tend to counterract the e f f e c t of the in te rna l costs i f they are important at a l l , which the small and s t a t i s t i c a l l y i n s i g n i f i c a n t parameter estimate on d^ K suggest they may not be. These r esu l t s are cons i s ten t with the imp l i ca t ions der ived from ana l ys i s o f the parameters, and again c a l l s in to question the v a l i d i t y of est imat ing costs of adjustment models based only on external c o s t s . I t a l so suggests that the e f f e c t s of monopsony in the cap i t a l market are n e g l i g i b l e . Since i t i s c l e a r that the external costs spec i f i c a i on i s s t a t i s t i c a l l y i n s i g n i f i c a n t and therefore may not provide add i t iona l useful in format ion, i t i s useful to consider what the adjustment costs funct ion looks l i k e in the absense of external cos t s . From Figure 3, der ived from the gross-internal s p e c i f i c a t i o n , i t appears that t h i s s p e c i f i c a t i o n r e f l e c t s tota l costs of -74-adjustment qui te w e l l , beginning at a p o s i t i v e leve l of cos ts which i s more p l aus ib l e than the previous s p e c i f i c a t i o n and then increas ing rap id l y at approximately the same rate as the general s p e c i f i c a t i o n . This model therefore appears to capture the important proper t ies of investment behavior . F igure 4 presents a representat ion of to ta l costs of adjustment fo r an external costs of adjustment only model, although t h i s s p e c i f i c a t i o n appears to be qui te c l e a r l y re jected based on parameter est imates. The diagram ind ica tes that without s p e c i f i c a t i o n of in terna l adjustment c o s t s , external costs r e f l e c t some of the pos i t i v e and convex costs of adjustment necessary fo r a p a r t i a l adjustment investment model. There are two important po ints to note about t h i s diagram, however. F i r s t , from the scale of the ve r t i c a l axis i t i s c l e a r that these costs are not increas ing nearly as r ap id l y as would be expected and i s character ized in the net investment model. This i s fu r ther h igh l ighted by the implaus ib ly high adjustment c o e f f i c i e n t (x | of .468 reported in Table 1-1. In add i t i on , although these costs r e f l e c t the convexity needed fo r t h i s model, the r esu l t s of the general model ind ica te that t h i s i s a m i s - s p e c i f i c a t i o n ; the external costs parameter appears to be merely "p i ck ing up" the costs properly assoc iated with the in terna l costs parameters which are i n co r r e c t l y excluded from th i s model. This suggests that although an external costs only model may provide reasonably p l aus ib l e est imates , i t i s a m i s spec i f i c a t i on and thus provides misleading impl i ca t ions as to the determinants of investment behavior. F i n a l l y , in te rna l cos ts of adjustment on net investment are p ic tured i n Table 5, beginning at a net investment leve l of zero , which i s c lose to the range of net investment in the l a te ' 7 0 ' s . This again appears pos i t i v e and convex and increases qu i te r a p i d l y , although from other parameter ind i ca t ions t h i s appears to be a r e l a t i v e l y uninformative s p e c i f i c a t i o n . -75-In summary, the model developed in th i s Essay appears to be capable of d i s t ingu i sh ing between the e f f e c t s of in terna l and external costs o f adjustment on gross investment. Ana lys is of these e f f e c t s both in terms of parameter estimates and graphs o f the funct ions ind ica te that in terna l costs of adjustment appear to be an important determinant of slow adjustment behavior, p a r t i c u l a r l y fo r gross investment, whereas external cos ts are of l i t t l e importance and may, in f a c t , s l i g h t l y counteract the e f f e c t s of in terna l costs of adjustment. The r e l a t i v e importance of in terna l cos ts i s based on three types of i n d i c a t i o n s , ( i ) the s t a t i s t i c a l s i g n i f i c a n c e , ( i i ) the numerical s i g n i f i c a n c e , i . e . , the to ta l magnitude of in terna l costs outweighs that of ex te rna l , and ( i i i ) the shape or convexity of the func t i on . Given these f ind ings I f i nd that the gross- interna l model i s my "p re fe r red " model, and w i l l emphasize resu l t s fo r t h i s model in the fo l lowing sec t ions , with occasional reference to comparable r e su l t s from other models. V .C . Monopoly I now turn to empir ica l cons iderat ion of another aspect of the model, the genera l i za t ion from per fec t competi t ion and cos t minimizat ion with exogenous output to a prof i t-maximizing monopoly s i t ua t i on where both output pr i ce and quantity are endogenous. The model presented e a r l i e r i s a lso capable of prov id ing various ind i ca to rs concerning the existence of imperfect competit ion in the output market, for example, the output demand e l a s t i c i t y , spec i f i c ind ices of monopoly power, output p r i ce e l a s t i c i t i e s , and pr i ce e l a s t i c i t i e s with respect to demand. I d iscuss these ind i ca to rs in tu rn . A basic imp l i ca t ion of a pe r fec t l y competi t ive output market i s that the output demand e l a s t i c i t i y aln Y/aln P i s i n f i n i t e . In the case of the -76-monopol is t , by con t ras t , t h i s demand e l a s t i c i t y i s an i nd i ca to r of monopoly power as i t r e f l e c t s the demand condi t ions fac ing the f i rm . Based on the estimated parameters discussed e a r l i e r , output demand e l a s t i c i t i e s are reported in Table 1-3 fo r var ious investment and costs of adjustment s p e c i f i c a t i o n s . For the "p re fe r red " gross-internal model the e l a s t i c i t y estimates are -3.167 in the short run, def ined as the time per iod where the monopolist responds by changing var iab le input demand and moves along i t s demand curve in response to current economic condi t ions and cap i t a l s tock, to -4.101 in the long run, def ined as the f u l l economic response as the monopolist changes i t s cap i t a l stock and experiences the f u l l e f f e c t s of i t s adjustments on the demand curve. These estimates are well above unity in absolute va lue , as i s required fo r the monopolist who must operate on the e l a s t i c por t ion of the demand curve. This e l a s t i c i t y i s reported fo r 1961, which i s the approximate mid-point of the sample, and which in add i t ion i s the year chosen for repor t ing e l a s t i c i t y estimates in many previous s tud ies , and therefore provides a useful bas is for comparison. Estimates for other years range from s l i g h t l y greater than unity to 5.0 in absolute va lue. For other models the e l a s t i c i t y estimates are s i m i l a r , although for the gross-external model the e l a s t i c i t y i s quite l a rge ; i t ranges to approximately -15.0 over the time pe r iod . Based on the prefer red s p e c i f i c a t i o n , these estimates are p l aus ib l e and provide support for the monopoly representat ion . Using the informat ion contained in th i s e l a s t i c i t y measure another t e s t o f the market s t ructure o f the industry can be der ived based on the idea that in the case of per fec t competit ion P=MC. In th i s model, such a r e s t r i c t i o n impl ies that B Y =0. An analogous r esu l t i s derived in Appelbaum (1977) in terms o f the inverse demand e l a s t i c i t y alnP/alnY = B Y / P . Using Appelbaum's framework, i f BY>0 and B Y / P <1 there i s consistency with c l a s s i c a l monopol is t ic -77-TABLE 1-3 Output Pr ice and Demand E l a s t i c i t i e s for A l t e rna t i ve Spec i f i c a t i ons on Investment and Adjustment Costs , 1948-77 (Reported fo r 1961) Gross Investment Gross Investment Internal and External Costs Internal Costs Only SR LR SR LR Output Demand E l a s t i c i t i e s e yp -2.976 -3.910 -3.167 -4.101 IL .336 .256 .316 .244 Output Pr ice E l a s t i c i t i e s epL .303 .476 .269 .308 e pE .017 .036 .025 .038 GpM .377 .636 .568 .619 GPPK 0 .020 0 .028 epK 0 -.673 0 -.901 Demand E l a s t i c i t i e s e L a 3.262 10.298 2.778 7.017 e Ea 2.754 4.562 4.109 6.363 eMa 4.120 5.623 7.288 7.791 £ Ka 0 1.831 0 3.218 e L , i n c .379 1.195 .850 2.147 G E , i n c .320 .529 1.257 1.947 E M, inc .478 .653 2.230 2.384 e K , i n c 0 .213 0 .985 -78-TABLE 1-3 (cont 'd) Output P r i ce and Demand E l a s t i c i t i e s fo r A l t e rna t i ve Spec i f i c a t i ons on Investment and Adjustment Costs , 1948-77 (Reported fo r 1961) Gross Investment Net Investment External Costs Only Internal Costs Only SR LR SR LR Output Demand E l a s t i c i t i e s e yp -4.776 -6.300 -3.168 -3.176 IL .209 .159 .366 .315 Output Pr ice E l a s t i c i t i e s e pL .259 .294 .288 .286 EpE .016 .031 .012 .028 epM .571 .603 .523 .533 EPPK 0 .029 0 .053 GpK 0 -.514 0 -.321 Demand E l a s t i c i t i e s e<x 8.213 10.796 3.799 3.235 eEK 4.302 7.862 2.252 4.075 EMa 10.844 11.173 6.696 6.620 eKK 0 5.483 0 4.332 E L , i n c 1.448 1.903 .606 .516 e E , i n c .759 1.386 .359 .650 eM , inc 1.912 1.990 1.067 1.055 G K , i n c 0 .967 0 .691 / -79-behavior, and i f BY>0 but B Y / P >1 fo r some time periods th i s impl ies mark-up monopol ist ic behavior. Th is can be framed a l t e rna t i v e l y wi th in more conventional theore t i ca l ana lys i s in terms of the Lerner index of monopoly power, a concentrat ion index commonly used in the i ndus t r i a l organizat ion l i t e r a t u r e . The Lerner index i s computed as ( P - M O / P and i s based on the notion that when per fect competit ion ex i s t s P = M C so that the index becomes ze ro , and when imperfect competit ion e x i s t s , P > M C so the r a t i o i s greater than zero . The c lose r the index i s to unity the more monopoly power the f i rm i s said to possess. It i s easy to see that t h i s r a t i o must equal the rec iproca l of the absolute value of the e l a s t i c i t y of demand, or the inverse demand e l a s t i c i t y of Appelbaum in dua l i t y terms. Th is Lerner measure, represented by 1^, i s reported in Table 1-3. It i s f a i r l y robust across a l t e rna t i ve costs of adjustment s p e c i f i c a t i o n s , approximately .25 to .30 , although fo r the external costs of adjustment model i t more c l o se l y approaches a pe r fec t l y competi t ive model. The numbers from the gross- interna l model ind ica te s i g n i f i c a n t monopoly power, and more short run than long run monopoly power, an i n t u i t i v e l y p l aus ib l e set of r e s u l t s . Note that the above index i s genera l ly assumed to be fo r a f i r m , whereas I have used aggregate data , i . e . , at the industry l e v e l . Hence these tes ts based on th i s i nd i ca to r may be i n v a l i d . In other words, a monopol ist ic industry i s being modeled rather than a monopol ist ic f i r m , and since an industry i n which f i rms are per fec t competitors might appear to be monopol ist ic at the industry l e v e l , i t i s questionable whether competi t ive behavior can be d is t ingu ished from monopol ist ic behavior in the above manner. Appelbaum, however, considers th i s question and makes i t c l e a r that t h i s i s not the case. He points out that i f a l l f i rms were compet i t i ve , t he i r pr i ces and (marginal) shadow pr i ces would be the same, so that t h i s would hold - 8 0 -i n aggregate. Thus, unless the ind iv idua l f i rms are non-competitive so that P^MC, monopol ist ic elements w i l l not be evident in the model. I.e., "the t e s t f o r B Y = 0 does not t e s t whether the demand curve fac ing the industry i s h o r i z o n t a l , but whether the dec is ion ru le i s to equate pr i ce with marginal II 44 c o s t . . . . i t t es t s whether or not B appears in the shadow pr i ce equat ion. Thus, i f one can perform a s t a t i s t i c a l t es t of monopoly power based on th i s r a t i o , inferences based on the t e s t w i l l be v a l i d . In t h i s case i t i s easy to see that to t e s t f o r per fec t competit ion or B Y = 0 , i t i s s u f f i c i e n t to t e s t whether B = 0 . For a s ing le r e s t r i c t i o n th i s can eas i l y be accomplished by looking at the " t " s t a t i s t i c . As the t s t a t i s t i c on s i s 8 . 8 1 fo r the gross- interna l s p e c i f i c a t i o n and i n no case i s below t h i s va lue, B appears to be s i g n i f i c a n t l y d i f f e r e n t from zero and the nu l l hypothesis of per fec t competit ion can be r e j ec ted . When th i s s i gn i f i c ance of B i s coupled with a Lerner index that c l e a r l y ind ica tes c l a s s i c a l monopoly behavior , there i s evidence i nd i ca t i ng that using models which assume per fec t l y competit ive behavior, p a r t i c u l a r l y as micro-foundations fo r measurement of macro ind i ca to rs such as product i v i t y and capac i ty u t i l i z a t i o n , may lead to erroneous r e s u l t s . Instead both output p r i ce and quanti ty should be viewed as endogenous. One imp l i ca t ion of t h i s i s that the numerous previous cost funct ion studies that have assumed output i s exogenous may therefore be subject to ser ious m i s spec i f i c a t i on er rors and econometric s imultanei ty b i ases . Another imp l i ca t ion of the important endogenous output quantity e f f e c t i s that the cha rac te r i za t ion of pr i ce e l a s t i c i t i e s of demand fo r inputs should a l low fo r output to ad jus t , rather than const ra in movements along given isoquants . Thus i t appears important in modeling the e f f e c t s of input p r i ce changes on t he i r quant i ty demanded, to c a l cu l a t e e l a s t i c i t i e s al lowing both the movement along an isoquant--the pure subs t i tu t ion e f fec t—and s h i f t s in isoquants—the sca le e f f e c t—bo th .o f which are important components of the -81-short run adjustment of the f i r m . The long run adjustment o f the f i rm then incorporates a lso the f u l l adjustment to the steady s t a te . As mentioned i n the previous sec t ion , I charac ter ize these as very short run (VSR), short run (SR), and long run (LR) e f f e c t s r e spec t i ve l y . These e f f e c t s can be represented by var ious e l a s t i c i t i e s . However, s ince so many var ious types of e l a s t i c i t i e s can be spec i f i ed with in th i s model, I emphasize only the main points b r i e f l y . The f i r s t set o f e l a s t i c i t i e s to cons ider are those most d i r e c t l y re l a ted to monopoly power as discussed above, which I c a l l output p r i ce e l a s t i c i t i e s and p r i ce e l a s t i c i t i e s with respect to demand, or demand e l a s t i c i t i e s . S p e c i f i c a l l y , these charac ter ize the f l e x i b i l i t y of the monopol is t ' s output p r i ce in response to exogenous shocks in va r i ab le input p r i c e s , and adjustments in var iab le input demand subject to s h i f t s in the exogenous demand funct ion fo r output, r e spec t i v e l y . Since they are def ined only in terms of the adjustment i n output demand v a r i a b l e s , I def ine them only f o r the short and long run. The output pr i ce e l a s t i c i t i e s are useful for i n te rp re t ing the importance of the v a r i a b i l i t y of output p r i ce as a dec is ion var iab le of the monopol ist . As can be seen in Table 1-3, the magnitudes of these e l a s t i c i t i e s are qui te subs t an t i a l , given the share of each input . The e f f e c t s of increases in the pr i ces of l abor , energy, mater ia ls and cap i t a l are a l l to increase output p r i c e . I n tu i t i v e l y i t would appear that t h i s i s cons is tent with p r o f i t maximizing behavior; an increase in the p r i ce of an input would increase the cos t of production and thus r e su l t in a decrease in output, corresponding to an increase i n the p r i ce of output. These e l a s t i c i t e s i nd i c a t e , fo r example, that an increase of one percent i n the pr i ce of labor impl ies a .30 percent increase in output p r i ce i n the short run--a substant ia l increase . By con t r a s t , a one percent increase in the pr i ce of energy causes output pr i ce to -82-increase by only .017 percent . Th is i s cons i s t en t , however, with these inputs ' r e l a t i v e impact on product ion ; as energy has a much smal ler share in costs a change in i t s p r i ce should have a smal ler to ta l impact on the p r i c i n g behavior of the f i rm . Note that these e l a s t i c i t i e s are everywhere la rger in the long run than the short run . Th is would appear to r e s u l t from the substant ia l returns to sca le impl ied by the estimated model. In the long run, as a l l i n t e r a c t i ng e f f e c t s work through the model, an increase in the pr i ce of a va r iab le input , say, labor , causes the scale of operat ions to decrease s u b s t a n t i a l l y , and thus des i red cap i t a l stock as well as demand fo r a l l va r iab le inputs to decrease. This i s impl ied by the la rger long run output p r i ce inc reases . I f the pr i ce of labor were increased, the f i rm des i res to decrease the scale of product ion , but i s constra ined by the level of the cap i t a l s tock, and therefore decreases production only p a r t i a l l y , decreasing var iab le inputs and output pr i ce toward f i n a l equ i l i b r i um. Note that t h i s decrease in var iab le input use can be in te rpre ted as a decrease i n u t i l i z a t o n o f the c ap i t a l stock even when the stock i t s e l f i s f i x e d . In the long run, the monopolist can a lso decrease i t s c ap i t a l stock to maximize p r o f i t s , fu r ther decreasing i t s output and increas ing output p r i ce along the demand curve. The output p r i ce therefore acts as an important dec i s ion var iab le in two ways; i t serves as a buf fer by a l lowing the f i rm to decrease i t s va r iab le inputs l ess than would otherwise be necessary to maximize short run p r o f i t s , thereby "smoothing" va r iab le input demand, and a lso allows the monopolist to cover some increas ing costs by s e l l i n g at a higher p r i c e . The signs of these c o e f f i c i e n t s thus co inc ide with a p r i o r i expectat ions and the magnitude ind ica tes that the f l e x i b i l i t y o f output p r i ce i s important for modeling the monopol is t ' s behavior. -83-Another i nd i ca to r of the f l e x i b i l i t y of output p r i ce i s the e l a s t i c i t y of output pr i ce with respect to a change in the leve l of the cap i t a l s tock, which provides impl i ca t ions about what the pattern of adjustment to long run equ i l ib r ium may look l i k e . Note that t h i s e l a s t i c i t y must be in te rpre ted d i f f e r e n t l y than the other e l a s t i c i t i e s ; i t determines the e f f e c t of short run f i x i t y of c ap i t a l on adjustment. The a n a l y s i s , however, i s analogous to above; s ince the monopolist i s able to decrease i t s c ap i t a l stock i t can fur ther increase output p r i ce given the negative c o e f f i c i e n t on e p K . Conversely, i f the monopolist wishes to increase i t s production but the cap i t a l stock i s a binding cons t ra in t i t w i l l keep i t s output p r i ce high in the short run to maximize p r o f i t s and then al low i t to decrease as i t adjusts the cap i t a l stock, gradual ly re lax ing the cons t r a i n t . The pr ice e l a s t i c i t i e s with respect to demand provide fur ther informat ion on the behavior of the monopol ist . These e l a s t i c i t i e s r e f l e c t changes in input demand with respect to changes in output demand, which i s represented by s h i f t s in the demand curve from e i the r changes in the in te rcept term (a) , or income (INC) in the demand curve s p e c i f i c a t i o n . The impl i ca t ions of these e l a s t i c i t i e s are s im i l a r to those f o r the output p r i ce e l a s t i c i t i e s d iscussed above. The demand e l a s t i c i t i e s suggest that given an increase in demand represented by an outward s h i f t o f the demand curve with a l l e lse f i x e d , va r i ab le input demand increases in the short run with cap i t a l f i x e d , but increases even fur ther when the f u l l increase in production and thus the sca le of operat ions in general i s not constra ined by the leve l of the cap i t a l s tock. A f i n a l important c h a r a c t e r i s t i c of the e l a s t i c i t y spec i f i c a t i ons in t h i s model i s the a b i l i t y to represent both components of the short run pr i ce e l a s t i c i t i e s fo r va r i ab le i n tpu t s , the pure subs t i tu t ion (output f ixed) and to ta l short run (output var iab le ) impacts of changing exogenous var iab les on var iab le input demand. These e l a s t i c i t i e s are reported as VSR and SR -84-e l a s t i c i t i e s in Table 1-4. From b r i e f inspect ion i t appears that the e l a s t i c i t i e s fo r the prefer red (gross- interna l ) s p e c i f i c a t i o n are in general i n t u i t i v e l y appeal ing, although some resu l t s within the other spec i f i c a t i ons are open to quest ion. S p e c i f i c a l l y , in the preferred s p e c i f i c a t i o n , a l l own e l a s t i c i t i e s are negative except fo r the very short run labor own e l a s t i c i t y . The output e f f e c t in t h i s case however causes the e l a s t i c i t y to take on a more i n t u i t i v e sign and magnitude. In add i t i on , a l l inputs appear to be long run complements, a r e s u l t of substant ia l returns to s ca l e . Th is i s cons is tent with the previous d iscuss ion of demand e l a s t i c i t i e s . For example, i f the p r i ce o f a va r i ab le input , say, l abor , increases , t h i s w i l l imply a decrease in overa l l scale and therefore a decrease in a l l other inputs and increase in the output p r i ce only i f t h i s overa l l complementarity e x i s t s . In add i t i on , the long run e f f e c t w i l l be la rger than the short run in th i s case , as the cap i t a l accumulation w i l l a l so r e f l e c t th i s complementary e f f e c t . Note that the output and output p r i ce v a r i a b i l i t y e f f e c t i n the short run appears to be quite important; estimates of the short run e l a s t i c i t i e s have a reasonable magnitude a well as s ign in a l l cases , and are subs tan t i a l l y 45 d i f f e r e n t than the very short run (pure subs t i tu t ion ) e f f e c t . Thus i t appears that the monopolist s p e c i f i c a t i o n i s a useful one fo r represent ing the behavior of f i rms in the manufacturing indust ry . The f l e x i b i l i t y of the monopol is t ' s dec is ion var iab les appears to capture behavioral cha r a c t e r i s t i c s that are important fo r modeling actual behavior, in cont ras t to the more r e s t r i c t i v e cos t funct ion model with exogenous output. -85-TABLE 1-4 P r i ce E l a s t i c i t y Estimates fo r A l te rna t i ve . Investment and Adjustment Cost S p e c i f i c a t i o n s , 1948-77 (Reported fo r 1961) Gross Investment Gross Investment Internal and External Costs Internal Costs Only VSR SR LR VSR SR LR e LL .111 -.415 -1.629 .091 -.243 -.275 e LE -.009 -.040 -.137 -.055 -.083 -.026 eLM -.102 -.782 -2.482 -.036 -.733 -.053 ELK 0 0 -.102 0 0 -.070 eEL -.056 -.517 -.896 -.312 -.799 -.762 e EE -.012 -.038 -.074 -.126 -.170 -.284 eEM .067 -.489 -1.219 .437 -.598 -2.443 eEK 0 0 -.041 0 0 -.078 eML -.047 -.729 -1.199 -.015 -.883 -.638 eME .005 -.033 -.088 .034 -.046 -.188 eMM .042 -.800 -1.722 -.018 -1.850 -3.156 eMK 0 0 -.046 0 0 -.094 <*L 0 0 -.390 0 0 -.381 eKE 0 0 -.030 0 0 -.055 eKM 0 0 -.439 0 0 -.858 eKK 0 0 -.027 0 0 -.031 -86-TABLE 1-4 (cont 'd) P r i ce E l a s t i c i t y Estimates fo r A l t e rna t i ve Investment and Adjustment Cost S p e c i f i c a t i o n s , 1948-77 (Reported fo r 1961) Gross Investment Net Investment External Costs Only Internal Costs Only VSR SR LR VSR SR LR G LL .123 -.811 -.630 .069 -.409 -.144 e LE -.0015 -.059 -.086 -.003 -.024 -.026 eLM -.121 -2.192 -1.978 -.066 -.932 -.399 ELK 0 0 -.124 0 0 -.091 G EL -.009 -.500 -.666 -.018 -.302 -.314 G EE -.060 -.092 -.132 -.056 -.072 -.076 GEM .069 -1.012 -2.115 .075 -.434 -.517 eEK 0 0 -.094 0 0 -.105 eML -.05 -1.290 -1.151 -.028 -.869 -.727 GME .005 -.071 -.154 .006 -.030 -.045 GMM .049 -2.683 -4.064 .022 -1.505 -2.005 GMK 0 0 -.140 0 0 -.111 GKL 0 0 -.647 0 0 -.537 eKE 0 0 -.070 0 0 -.060 EKM 0 0 -1.428 0 0 -.731 eKK 0 0 -.056 0 0 -.165 VI. Concluding Remarks -87-Since so many d i f f e r e n t issues have been d iscussed , in conc lus ion I b r i e f l y summarize the main propos i t ions and f ind ings contained in th i s essay. The p r inc ipa l argument of th i s essay i s that ex i s t i ng dynamic models of f ac to r demand, although incorporat ing many extensions over previous work which f a c i l i t a t e the i r use for exp la in ing the behavior of the U.S. manufacturing indus t ry , re ta in some assumptions that are too r e s t r i c t i v e to r e f l e c t the many dec is ion var iab les the f i rm must take into cons ide ra t ion . Extensions over previous work which I have emphasized here inc lude : ( i ) cons iderat ion of both interna l and external adjustment cost components, each of which could imply slow investment behavior fo r the f i rm , but whose importance i s an empir ica l issue that should be pursued, ( i i ) cons iderat ion of the v a l i d i t y of spec i f y ing adjustment costs on gross vs . net investment, and ( i i i ) extension to an imperfect ly competi t ive f i rm s p e c i f i c a t i o n which incorporates endogenous output and output p r i ce dec is ions subject to an exogenously given downward s lop ing demand curve. Based on est imat ion of a system of demand equations for a monopol ist ic f i rm which incorporates the f l e x i b i l i t y necessary for der i v ing tes ts of these quest ions, I reached several tenta t i ve conc lus ions : ( i ) The model appears to r e f l e c t behavior in the U.S. manufacturing sector f a i r l y we l l , e spec i a l l y fo r a model assuming in terna l costs only on gross investment, ( i i ) Internal adjustment costs appear both s i g n i f i c a n t , numerical ly and s t a t i s t i c a l l y , and a lso have the " co r rec t " shape to induce slow cap i ta l accumulation fo r the f i r m , as compared to external cos t s . F i n a l l y , ( i i i ) the monopoly s p e c i f i c a t i o n , s p e c i f i c a l l y the f l e x i b i l i t y of output p r i ce (and thus output) adds a dec is ion var iab le or "bu f f e r " fo r the f i rm in response to exogenous shocks. This appears important for represent ing the manufacturing indus t ry , -88-and was demonstrated using concentrat ion i n d i c e s , output p r i ce e l a s t i c i t i e s , demand e l a s t i c i t i e s , and the d i f f e rence between the pure subs t i t u t i on and to ta l output adjusted e f f e c t of exogenous changes on input demand. Although the r e su l t s presented here accord well with the conc lus ion that t h i s general model has the a b i l i t y to represent important c h a r a c t e r i s t i c s of f i rm behavior , many other fur ther refinements and genera l iza t ions are a lso important to take into cons ide ra t ion . Some of these extensions which I hope to pursue in future research are discussed in the conc lus ion to t h i s d i s s e r t a t i o n . The one which I fee l i s the most important, however, the extension to non-stat ic expectat ions on r e l a t i v e input p r i ces and p a r t i c u l a r l y output, i s undertaken in the fo l lowing essay. -89-Footnotes 1 See Treadway (1969, 1971, 1974), Lucas (1967, 1970), Gould (1968), Brechl ing (1975), and Mortensen (1973). 2 See Nickel 1 (1978). 3 See Nickel 1 (1978). The importance of Assumptions 1 and 2 i s that they al low one to compare income flows and stocks of money at d i f f e r e n t points in time by c a l c u l a t i n g t he i r present va lue. A lso the u t i l i t y maximization of the f i r m ' s owners i s then cons i s ten t with maximization of the present value of the income stream of the f i rm . As long as there i s per fect c e r t a i n t y , one can separate the consumption and investment dec is ions of the owners. 4 For a d iscuss ion of u(t) see Nickel 1 (1978), ch . 2. 5 S p e c i f i c a l l y , i f adjustment costs are given by C(Z ) , we have C'(Z)>0 as Z>0, C"(Z)>0. 6 N i cke l l (1978), ch . 3. 7 N icke l l (1978). 8 Rothschi ld (1971). 9 Gould (1968). 1 0 Dynamic adjustment mechanisms other than the f l e x i b l e acce le ra tor form can be generated from the assumptions of adjustment c o s t s , e . g . , hump-shaped d i s t r i b u t i o n s are poss ib le and can be generated from some funct ional forms (Epstein (1979)). H The endogeneity of t h i s adjustment process i s in d i r e c t contras t to ad hoc pa r t i a l adjustment models, as adjustment depends on the parameters of the system and var ies with r. However, i f G were quadrat ic so that G Xx a n d G X x were constant parameters, and i f r were r e l a t i v e l y s t ab le , x 2 i would a lso tend to be quite s t ab le . 1 2 Using s im i l a r model l ing methods, t h i s approach has been pursued by, among others , Gould (1968), N icke l l (1977), and Epstein (1979). 1 3 Refer to N i cke l l (1978, ch . 5) for an e laborat ion of these po in t s . 1 4 Nerlove (1972). 1 5 Or varying l eve l s of other buf fer s tocks , such as inventor ies (see Caves et a l . , (1979)). 1 6 See Fe lds te in and Rothschi ld (1974). 1 7 See N i cke l l (1978), Rothschi ld (1971), Gould (1968), Mussa (1977). 1 8 See Diewert (1978). Note a lso that the inverse demand funct ion D(Y) must s a t i s f y ce r t a in requirements. S p e c i f i c a l l y , Y i s assumed to be p o s i t i v e , D(Y) i s d i f f e r e n t i a t e at Y>0, and D'(Y) 0. -90-1 9 This formulat ion r e l i e s heavi ly on Diewert (1978), who developed a s im i l a r precedure in the s t a t i c context . 20 In other words, we do not have an "OPEC-effect" (see Robert Pindyck (1979)), where i t may be optimal for the producer to have a sudden increase in p r i ce because demand i s more i n e l a s t i c in the short run than in the long run. 21 This could occur as long as these changes in u t i l i z a t i o n were a lso poss ib le c o s t l e s s l y and instantaneously . 22 i t might be useful to explore models that impose increas ing marginal costs o f adjustment fo r output p r i ce and/or other adjustments l i k e inven to r i es , capac i t y , or other buf fer s tocks , thereby smoothing output p r i ce var ia tons . 23 Note a lso that the second order condi t ions are s a t i s f i e d by the imposi t ion of normal r egu l a r i t y cond i t ions on the production func t i on . 24 The norma l i zed . res t r i c t ed cost func t ion , based on a production funct ion with the usual p rope r t i e s , has been shown by Lau (1976) to have the fo l lowing p rope r t i e s : ( i ) G i s decreasing in x, and increas ing in P, Y and x, i e : G x<0, G p O , Gp>0, and Gy>0. ( i i ) G i s concave in P, convex in x, and convex in x. ( i i i ) Gpj=Vj=the cond i t iona l cost minimizing input l e v e l . 25 in add i t i on , expectat ions cons iderat ions and lags o f adjustment are c l o se l y in ter tw ined. S e n s i t i v i t y o f d i s t r i bu t ed lag estimates may depend on the process of expectat ion formation (Nadir i and Rosen (1974)), i . e . , estimated adjustment c o e f f i c i e n t s may capture e r rors in fo recas t ing exogenous var iab les as well as genuine adjustment cos t s . 26 See Mortensen (1973) for a more complete d iscuss ion of t h i s cons t r a in t . 27 The technology represented by t h i s cos t funct ion i s based on a non-constant returns to sca le production func t ion . Constant returns to sca le (CRTS) can be imposed on the model (Morrison-Berndt (1981)), but CRTS i s not a spec ia l parametric case of the non-constant returns formulat ion . For the monopoly model, i t seems more reasonable to assume non-constant re turns . It i s important to r e a l i z e , however, that even with CRTS in th i s model, a de te rmin is t i c output leve l r e su l t s under p r o f i t maximization, both because inc reas ing marginal costs o f investment are on gross rather than net investment (Gould (1968)), and because of the monopol ist ic assumption (Nickel 1 (1978)). In other words, with imperfect competit ion in e i the r the output and/or fac to r market the CRTS model has a de te rmin is t i c output. 28 Note that the r esu l t s may depend on which input i s considered to be the numeraire, as the demand funct ions for the var iab le inputs are not symmetric. This could be a major problem with empir ical implementation. In order to check t h i s i t would be useful to cons ider a l t e rna t i ve var iab le inputs as the numeraire and tes t for s e n s i t i v i t y of the est imates. I f the estimates were very sens i t i v e to the s p e c i f i c a t i o n , a more complex but symmetric model der ived from a va r i ab le p r o f i t funct ion may be p re fe rab le . - 9 1 -2 9 A c l o s e l y re la ted s p e c i f i c a t i o n can be developed by d i v i d i ng L in to production (unsk i l l ed or blue c o l l a r , B) and non-production ( s k i l l e d or white c o l l a r , W) workers and a l lowing W to be a second quas i-f ixed input with a labor-accumulation equation equiva lent to ( 1 . 3 . 3 2 ) . In such a s p e c i f i c a t i o n , the normal iz ing input could become B instead of L . This approach incorporates a labor-hoarding concept s ince the model could e x p l i c i t l y al low fo r " turnover" costs of adjustment. Investment in th i s quas i-f ixed input must be net instead of gross , however, as labor turnover cannot be taken into account due to data l i m i t a t i o n s . 30 Ideal ly empir ica l implementation using more time-disaggregated data , say quar te r l y , would be p re fe rab le . Data l im i t a t i ons prevent t h i s , however, as adequate information on E and M disaggregated more completely for time or industry seem to be unava i l ab le . In a d d i t i o n , quarter ly cap i t a l stock data tends to be simply in te rpo la ted yea r l y data , which would incorporate an add i t iona l e r ro r in to an already econometr ica l ly complex model i f u t i l i z e d for es t imat ion . It appears, the re fore , that the annual level of aggregation i s the most j u s t i f i a b l e fo r empir ica l implementation of the model given the ex i s t i ng cons t ra in ts on a v a i l a b i l i t y of data . 3 1 G r i l i c he s ( 1 9 6 7 ) . 3 2 Grether and Maddala ( 1 9 7 3 ) suggest tha t : "In the case of d i s c re te models with no lags or i n f i n i t e l ags , measurement e r rors may lead to the spurious appearance of long adjustment lags even in large samples. This i s because aggregation over time induces a pos i t i v e dependence between aggregated time disturbances and the lagged values of the aggregate dependent var iab le and causes us to overestimate the impl ied average l a g s . " (Note a lso that the notion of returns to sca le i s time dependent.) Of course, with continuous models and d i s c re te representat ions , some er rors of measurement are i n e v i t a b l e . It i s therefore not su rp r i s i ng that estimates based on annual data often imply longer lags and thus a longer time-dependence than s im i l a r estimates based on quar ter ly data . Since t h i s long lag s t ructure may not be t r u l y s i g n i f i c a n t , even i f the true l ag i s r e l a t i v e l y short fu r ther problems may r e su l t i f the lag i s a r b i t r a r i l y t runcated. 3 3 Note that here I only consider systems est imation techniques which are asymptot ica l ly e f f i c i e n t as well as cons i s t en t , in contras t to s ing le equation est imat ion which disregards the co r r e l a t i on of the disturbances across equations or p r i o r r e s t r i c t i o n s on other equations in the model, and therefore neglects re levant information which allows for more e f f i c i e n t es t imat ion . 3 4 See Hansen and Sargent ( 1 9 7 9 , 1 9 8 0 ) . 3 ^ I f the two r e l a t i v e pr i ces are a lso considered to be endogenous, an add i t iona l two excluded exogenous var iab les are necessary. In the f i n a l es t imat ion , however, I found that incorpora t ing add i t iona l exogenous var iab les in the system appeared to have neg l i g i b l e e f f e c t on the r e s u l t s . I therefore reta ined the s p e c i f i c a t i o n with only three excluded exogenous var iab les in the demand equat ion. 3*> In many cases i t was not poss ib le to determine the v a l i d i t y of the cons t ra in t on y^, however, as the existence of t h i s extra term in the adjustment parameter caused the term " i n s i d e " the square root to become negative so that no r e su l t s could be obta ined. I t appears that the r a t i o -92-within the square root s ign becomes very v o l a t i l e with too many f ree parameters, causing convergence problems. Note a lso that Y Y K was in general very i n s i g n i f i c a n t ; in the general gross investment s p e c i f i c a t i o n with in terna l and external cos t s , for example, re lax ing the r e s t r i c t i o n on Y Y K y i e l ded a t - s t a t i s t i c of .386. 3 7 Note that using only the 1948-71 data , however, t h i s hypothesis ^s r e j e c ted ; the t s t a t i s t i c on d|<K on Table 1-2 i s -3.085. 38 A j o i n t t es t in the case of i t e r a t i v e NL3SLS requires a Wald tes t because there i s no d i s t r i bu ton theory a l lowing for other types of hypothesis t e s t i n g . Carry ing out t h i s t es t of the f i ve PR r e s t r i c t i o n s r esu l t s in a Chi-square s t a t i s t i c fo r a 5 percent s i gn i f i c ance leve l of 6.29 as compared to the c r i t i c a l value of 11.07. 3 9 In the 1948-71 s p e c i f i c a t i o n the c o e f f i c i e n t i s not only negative but s t a t i s t i c a l l y s i g n i f i c a n t , a l lowing for a more v a l i d asser t ion of concav i t y . 4 0 See E isner-St rotz (1963). 4 0 The existence of economies of sca le does not make sense with pe r f ec t l y competit ive supp l i e s . With monopoly i t i s not necessar i l y the case that s h i f t s in the demand curve imply decreases in the pr i ce unless returns to sca le are very h igh. 41 Note, however, that for the 1948-71 time per iod , which both BFW and MB, as well as PR emphasize, the net investment s e p c i f i c a t i o n with in terna l costs appears to be more appropriate according to the R 2 and adjustment c o e f f i c i e n t est imates. 4 2 This ind ica tes that the costs of adjustment model i s v a l i d fo r represent ing slow adjustment behavior ; a n a l y t i c a l l y t h i s impl ies that the r e s t r i c t i o n s for the adjustment parameter to be pos i t i v e and less than unity ho ld . 4 3 Appelbaum, p. 287. 4 4 One apparent problem appears to be the change in sign of the mater ia ls own-elast ic i ty between the VSR and SR, as an important p r i n c i p l e o f production theory i s that when both subs t i tu t ion and output or scale e f f e c t s e x i s t , the output e f f e c t must r e in fo rce the o r i g i na l own-substitution e f f e c t ; there are no G i f f en goods in production (Nagatani (1978)). This apparent con t rad i c t i on does not ho ld , however, s ince th i s p a r t i c u l a r statement of the p r i n c i p l e only holds when the subs t i tu t i on e f f e c t i s negat ive, which by economic theory i t must be. Thus th i s r e s u l t i s cons is tent with the p r i n c i p l e that the output e f f e c t i s negat ive; in t h i s case i t overr ides the i n i t i a l erroneous subs t i tu t i on e f f e c t . -93-Essay 2. A St ructura l Model of Dynamic Factor Demands with Non-Static Expectations I. Introduct ion Unt i l recent ly very l i t t l e l i t e r a t u r e ex is ted on incorporat ion of non-stat ic pr ice and output expectat ions into empi r i ca l l y implementable f ac to r demand models. The vast l i t e r a t u r e on f l e x i b l e funct ional forms can almost be completely charac ter ized by s t a t i c expectat ions on pr i ces and output as well as instantaneous adjustment to des i red l e v e l s of demand. More recent studies have incorporated inc reas ing ly complex theore t i ca l and econometric s t ruc tu res . The l i t e r a t u r e along the l i n e s of Fuss (1976) and Berndt, Fuss and Waverman (1979) (BFW) incorporates endogenous determination o f a dynamic adjustment s t ructure but re ta ins a s t a t i c expectat ions assumption. On the other hand, most current " r a t iona l expectat ions" models, fo l lowing the lead of Thomas Sargent and Robert Lucas, spec i fy a costs o f adjustment model but make very r e s t r i c t i v e s imp l i f y ing assumptions on the determin is t i c economic s t ructure in order r igorous ly to spec i fy the complex s tochast ic s t ructure impl ied by the ra t iona l expectat ions assumption. This essay provides an example of the former approach, emphasizing the s t ruc tura l model s p e c i f i c a t i o n , but i t a l so incorporates considerable a t tent ion to s tochast i c cons ide ra t ions . In i t I genera l ize the ' s t a te of the a r t ' BFW model out l ined in Essay 1 which i s based on the assumptions of cost-minimizing behavior by the f i r m , given s t a t i c expectat ions, per fec t compet i t ion, and in terna l costs of adjustment fo r net investment, to an analogous framework incorporat ing non-stat ic expectations about future output and ( r e l a t i ve ) pr i ces into the s t ruc tura l model. Th is i s based on the -94-assumption that the f i rm optimizes subject to a known future sequence of exogenous output demand l eve l s " J Y f t + s ^ Q and input p r i ce l eve l s for va r iab le |Pj(t+s)| s =Q and quas i-f ixed ja..(t+s)| s_ 0 i n p u t s . 1 Two issues must be addressed in order to formal ize and implement t h i s non-stat ic expectat ions extens ion. The f i r s t i s to determine the impact of a non-constant path of future exogenous var iab les on the decision-making process o f the f i r m . The non-constant future path of exogenous var iab les r e su l t s in varying " ta rge t " stocks fo r the f i rm over t ime, and therefore en t a i l s a more complicated determin is t i c time s t ructure than do ex i s t i ng models. The second issue i s to model the f i r m ' s formation of expectat ions about the path of future va r i ab l es . Th is could be accomplished by postu la t ing var ious models of expectat ion format ion, ranging from simple ext rapo la t i ve paths to adaptive and f i n a l l y to f u l l y " r a t i o n a l " models, in both determin is t i c and s tochast i c (uncertain) cases . Emphasis in t h i s essay i s on incorporat ion of adaptive and " p a r t i a l r a t i o n a l i t y " expectat ions assumptions into the s t ruc tura l model. This Essay represents , the re fo re , a step toward synthesis of the two primary approaches to dynamic input demand modell ing mentioned above, using approximations to ra t iona l expectat ions and ce r ta in t y equivalence to incorporate ant i c ipa tory behavior of f irms without s a c r i f i c i n g the r i ch economic s t ructure of the determin is t i c model. I emphasize the determin is t i c economic s t ructure rather than the soph is t i ca ted econometric techniques or s tochast i c spec i f i c a t i ons that t y p i c a l l y appear in the ra t iona l expectat ions l i t e r a t u r e . I bel ieve th i s approach al lows me to incorporate important components of the " r a t iona l expectat ions" l i t e r a t u r e , while re ta in ing s u f f i c i e n t s imp l i f y ing structure on expectat ions so that the economic s t ructure i s not neglected as a r e su l t of s t r ingent s i m p l i f i c a t i o n s required fo r the t r a c t a b i l i t y of the s tochast i c model. -95-The Essay i s s t ructured as fo l lows . In the next sect ion the e x i s t i n g l i t e r a t u r e on the two fundamental approaches to dynamic input demand model l ing i s d i scussed . Some basic conclus ions are : ( i ) a synthesis i s important, for the two primary approaches to th i s problem incorporate d i f f e r e n t r e s t r i c t i v e assumptions which can be genera l i zed ; ( i i ) the basic idea under ly ing the ra t iona l expectat ions hypothesis (REH) i s useful and important, but approximations to the "pure" REH idea may ac tua l l y be more useful emp i r i c a l l y for modeling behavior ; current charac te r iza t ions of " r a t i o n a l i t y " are too r e s t r i c t i v e ; and ( i i i ) time se r i es ana lys i s (TSA) can be a useful tool fo r spec i f y ing expectat ions formation for i t can charac ter ize a s i m p l i f i e d vers ion of the REH based on known past values of the same exogenous v a r i ab l e s , and can provide the bas is for t e s t ing other more simple and often used "approximations" to REH such as "adapt ive" expectat ions . The de te rmin is t i c model i s then der ived in Sect ion III, based on the e x i s t i n g BFW investment demand model but incorporat ing a moving " t a rge t " cap i t a l stock level depending on future paths of exogenous va r i ab l e s . The econometric implementation of the model i s then discussed in Sect ion IV, based on a " d i s t r i b u t e d lead" framework r e su l t i ng from a Koyck transform on an i n f i n i t e se r i es of future expected va lues . Th is transform f a c i l i t a t e s est imat ion based on known values of the exogenous va r i ab l e s . In Sect ion V I report the major empir ica l r e su l t s based on est imat ion using annual U.S. manufacturing data for 1948-77. A major f ind ing i s that the adaptive expectat ions representat ion appears to be an a t t r a c t i v e expectat ions s p e c i f i c a t i o n , f o r i t r e f l e c t s the important aspects o f non-stat ic expectat ions formation while r e t a in ing enough st ructure to i den t i f y the technologica l parameters. In the f i na l sect ion I present concluding remarks. II. Review of the L i t e ra tu re -96-II.A. Dynamics "Although a theory of dynamic economics i s s t i l l a th ing of the fu tu re , we must not be s a t i s f i e d with the status quo in s t a t i s t i c a l research. Many of the s t a t i s t i c a l researches that are c a r r i ed on in the soc ia l sc iences lack the i n sp i r a t i on of any theory—sta t i c or dynamic. Re la t i ve l y few of us l i k e to perform the necessary mental experiments which should precede the assembling and manipulating of the data i f the r e su l t s are to be s i g n i f i c a n t . . . . Research i s not good simply because i t i s mathematical or s t a t i s t i c a l , or because i t makes use of ingenious machines. Research i s good i f i t i s s i g n i f i c a n t , i f i t i s f r u i t f u l , i f i t i s cons is tent with es tab l i shed p r i n c i p l e s , or i f i t helps to overthrow erroneous p r i n c i p l e s . " 2 Since the time of H. Schul tz (1938), dynamic theory has evolved a great deal in terms of l i nk i ng dynamic theore t i ca l models with econometric measurement. Two c ru c i a l steps based on the "mutually supportive ro les of theory and measurement in economics" have been the development of a dynamic theore t i ca l s t ructure fo r investment and the r esu l t i ng theore t i ca l and econometric re l i ance on lag s t ruc tu res . Stock adjustment behavior and/or expectat ions formation have provided the conceptual framework fo r t h i s development. Most ear ly dynamic studies were based on simple adaptive expectat ions or pa r t i a l adjustment frameworks imposed on s t a t i c models. These models resu l ted in s p e c i f i c a t i o n s that r e l i e d on i n f i n i t e d i s t r i bu t ed l ags , transformed by a Koyck adjustment process to a t rac tab le s p e c i f i c a t i o n based only on current var iab les and a lagged dependent v a r i ab l e . Although at times they have served well in i nd i ca t ing the importance of the expectat ions and stock re la ted nature o f the dynamic behavioral process, these models have not been derived from a 3 r igorous theore t i ca l s t ructure and are e s s e n t i a l l y ad hoc . The pa r t i a l adjustment models tend to r e l y i m p l i c i t l y on costs of adjustment or de l i very lags which force slow adjustment to the "des i r ed " cap i t a l stock. The -97-re su l t i ng lag structure ' imposed on an e s s e n t i a l l y s t a t i c s t ructure charac te r izes "ad hoc" dynamics. S i m i l a r l y , the expectat ions formation theory has genera l ly been based on a simple adaptive expectat ions model with geometr ica l ly dec l in ing weights. Although i t has been shown that fo r a c e r t a i n form of nonstationary process th i s formation i s optimal (Muth (1961)), i t s t i l l requires theore t i ca l foundat ions. Development of a more complete s t ruc tura l opt imizat ion model of the f i r m ' s behavior as a basis fo r these 4 models i s an important extension toward a useful dynamic theory of the f i r m . In order to const ruc t a t ru l y dynamic model, i t i s necessary to spec i fy the laws of motion fo r both the stocks of cap i t a l and expectat ional v a r i ab l e s . These components of the f i r m ' s decision-making process are c l o se l y inter twined.^ If f o r some spec i f i c reason cap i t a l (or any other input such as s k i l l e d labor or inventor ies ) adjusts slowly due to convex costs of adjustment, i r r e v e r s i b i l i t y of investment and uncer ta in ty , or de l i very l ags , then a time s t ructure i s imposed on investment dec i s i ons , and expectat ions of future exogenous var iab les such as pr i ces w i l l be c ruc i a l f o r determination of cur rent changes in investment. By con t r a s t , i f a l l inputs can be var ied instantaneously and c o s t l e s s l y , myopic expectat ion behavior i s s u f f i c i e n t because only current values matter. This i n t e r r e l a t i onsh ip between slow adjustment of stocks and expectat ions formation has been emphasized by Arrow (1978): "Economic dec is ions are seen as mostly concerned with dec is ions on holding of assets rather than on choices of f lows. . . . Suppl ies of assets are l a rge l y constra ined by the past and only changable incrementa l ly . Decis ions to hold assets and to acquire increments are determined by the future o r , more p r e c i s e l y , by an t i c i pa t i ons of the f u tu r e . " (p. 157) In the past decade numerous dynamic investment researchers have proposed incorporat ing various aspects of "opt ima l " behavior of economic agents that r e s u l t in lag or lead s t ructures generated from costs of adjustment (stock -98-behavior) and/or informational cons t ra in ts (expectat ional behav ior ) . These formulat ions are important extensions toward putt ing addi t iona l s t ructure on both the stock and expectat ional e f f e c t s on investment, but due to complexity they ra re ly take into account both a r igorous s t ruc tura l model of investment dec is ions and optimal expectat ions formation behavior. Unfortunate ly , such complexity has required research in th i s area to expand in d i r e c t i ons which neglect the "mutually supportive ro les of theory and measurement" (Nerlove (1972)) in favor of pursuing t rac tab le cha rac te r i za t ions of one framework or the other . S p e c i f i c a l l y , much recent research has been based on formulat ion of complex representat ions of the s tochast ic s t ructure of dynamic models. In i t s most soph is t i ca ted forms th i s " t ime-ser ies " based approach has been presented in var ious papers by Thomas Sargent and others . The advantage of such an approach i s that "by formulat ing and est imat ing models which d iscr iminate between e f f e c t s of s t ruc tura l parameters of the object ive funct ional and the cons t r a i n t s , and the e f f e c t s of parameters descr ib ing the evo lut ion of exogenous va r i ab l e s , these authors are not subject to Lucas' (1975) c r i t i q u e of ad hoc es t imat ion . "^ There are a lso many disadvantages to th i s approach. For example, f ac to r demand models based on a s tochast ic maximum problem which in turn depends on time invar i an t parameters r e s u l t in an over ly s i m p l i s t i c theore t i ca l model of f i rm behavior. However, because of the complexity of the econometric aspects o f the problem, th i s s t ruc tura l s p e c i f i c a t i o n i s necessary, for ce r t a in t y equivalence i s required to s impl i f y the cons t ra in ts imposed by the ra t iona l expectat ions assumptions. In add i t i on , only a l im i t ed amount of s tochast ic s t ructure can be generated from ana lys i s of the f i n i t e amount of time ser ies data normally used to estimate any of these models. This i s recognized by Meese (1980) who f inds that meaningful r e su l t s cannot be obtained once one -99-expands Sargent 's one-factor model of labor demand to inc lude determination of c ap i t a l accumulation. At the other extreme are the Lucas-Treadway-type models which are based on f a i r l y complex de te rmin i s t i c theory of the dynamic investment behavior of f i rms , but which r e l y on r e s t r i c t i v e assumptions of expectat ions and market s t ructure in order to keep the models a n a l y t i c a l l y and emp i r i ca l l y t r a c t ab l e . These studies e x p l i c i t l y formulate a dynamic opt imizat ion cha rac te r i za t i on o f the f i r m ' s behavior and der ive a system of instantaneous demand and accumulation equations for va r i ab le and quas i-f ixed inputs , r e spec t i v e l y , based on an endogenous f l e x i b l e acce l e ra to r . These models were discussed in Essay 1. The theore t i ca l framework for such models i s r igorous , but the s tochast i c s t ructure i s t y p i c a l l y very s i m p l i s t i c . In p a r t i c u l a r , expectat ions formation i s assumed to be s t a t i c and c e r t a i n , so that any e x p l i c i t d i s t r i bu t ed lag formation e x p l i c i t l y generated from a s tochas t i c s t ructure based on expectat ions formation over time i s assumed away. Incorporat ion of a p r i o r i important determinants of economic behavior such as expectat ions into e x i s t i n g de te rmin is t i c models i s c l e a r l y important, but i t i s a lso ev ident that other equal ly important de te rmin is t i c theore t i ca l s t ructures should not be ignored in the process of pursuing such a goa l . The problem i s that both types of models qu ick ly become so complex with genera l i za t ion of t he i r r e s t r i c t i v e assumptions that a complete in tegra t ion i s i n t r ac tab l e a n a l y t i c a l l y and e m p i r i c a l l y . Using some judgment as to which parts of these models can be s i m p l i f i e d , however, I attempt here to provide d i r e c t i on towards a pa r t i a l synthes is . S p e c i f i c a l l y , in t h i s Essay I use the theore t i ca l framework of Treadway and incorporate what I consider to be the most important aspects o f expectat ions formation in to the model. Thus the problem can be broken down in to the two parts mentioned in the i n t roduc t i on ; the representat ion of the f i r m ' s dec is ion processes given a non-stat ic c e r t a i n -100-future path of exogenous v a r i ab l e s , and the s p e c i f i c a t i o n of the expectat ions process i t s e l f . I expand on these two issues separately in the fo l lowing two sub-sect ions. II.B. The F i rm's Investment Decis ions with Non-Static Expectat ions Most ex i s t i ng dynamic investment models in the l i t e r a t u r e formulate f ac to r demand models for the f i rm fac ing increas ing costs of adjust ing quas i-f ixed inputs , and der ive a d i s t r i bu t ed lag form for investment based on a p a r t i a l adjustment mechanism. Many authors have c r i t i c i z e d the use of t h i s " f i x e d ta rget " type of equ i l ib r ium model fo r econometric app l i c a t i ons , e spec i a l l y s ince f i rms have been observed to react to moving targets dependent on changing economic cond i t ions^ . A n a l y t i c a l l y , the property o f the so lu t ion as an adjustment towards a long run equ i l ib r ium des i red stock leve l depends c r u c i a l l y on the expectat ions assumption incorporated in to the model; i t requires " s t a t i c " expectat ions . In genera l , the optimal path over the en t i re fu tu re , inc lud ing the current time pe r iod , depends on the expected future path of exogenous var iab les such as p r i c e s . Even models that assume ce r t a in t y of t h i s future path but re lax the assumption of a constant path may be too complex to implement econometr ica l ly unless a substant ia l s imp l i f y ing s t ructure i s imposed. However, as Marc Nerlove has demonstrated, i f these future p r i ces are not assumed to be c e r t a i n , e i the r f i xed or changing, "the problem of der i v ing an econometr ica l ly useful cha rac te r i za t i on of optimal behavior under dynamic condi t ions becomes almost i n s o l u b l e ; indeed, i t i s not in general poss ib le to charac ter ize the 8 optimal paths a n a l y t i c a l l y . " Th is problem can be charac te r ized more forma l l y . Given a s ing le quas i- f ixed input c a p i t a l , a simple constant pr ice and output assumption -101-allows one to def ine the des i red c ap i t a l stock to be that quant i ty of c ap i t a l which the f i rm would employ i n long run equ i l ib r ium at these constant p r i c e s . In such a case , with the f l e x i b l e acce le ra tor mechanism the rate of accumulation of cap i t a l stock i s d i r e c t l y proport ional to the d i f f e rence between the des i red cap i t a l stock and the current c ap i t a l s tock : 2.2.1) K(t) = x(K* - K ( t ) ) , where K* i s a f i xed ta rget given s t a t i c expectat ions. However, i f exogenous var iab les such as pr ices are changing or are expected to change, the "des i r ed " c ap i t a l stock var ies over time and thus the f l e x i b l e acce le ra tor mechanism can be charac te r ized as : 2.2.2) K(t) = x(K*(t) + J - K(t ) ) = x( f (K* ( t ) ) - K(t ) ) = x(K**(t) - K ( t ) ) , where J represents the investment corresponding to the expected changes in future exogenous va r i ab l es . The problem i s that there r e a l l y i s no reason that K*(t) should be considered " d e s i r a b l e " ; i t i s simply the cap i t a l stock leve l that generates replacement investment such that the marginal adjustment cos t j u s t balances the current marginal net ga in . Here the " ta rge t " stock i s not very well de f ined . Th i s problem of spec i f y ing a "des i r ed " cap i t a l stock with var iab le exogenous var iab les provides one reason, along with the complexity of more r e a l i s t i c models, that most cur rent work has incorporated constancy of g exogenous var iab les or s t a t i c expectat ions . However, s im i l a r to Nickel 1 (1977), by def in ing K*(t) as that leve l of K that would be demanded in a steady state where pr ices and output are constant at time t l e v e l s , i t i s poss ib le a n a l y t i c a l l y to def ine a target c ap i t a l stock K**(t) as a convex combination of a l l the d i f f e r e n t "des i r ed " cap i t a l stocks K*(s) corresponding to the path of future var iab les expected with c e r t a i n t y . Within such a -102-s p e c i f i c a t i o n , the f l e x i b l e acce le ra tor formula po ten t i a l l y becomes subs tan t i a l l y more complex; the formulat ion spec i f i ed in (2.1.2) can be r e t a ined , but the ana l y t i ca l representat ion of K**(t) becomes cumbersome and thus less a t t r a c t i v e fo r empir ica l implementation. Given ce r t a in s imp l i f y ing assumptions about the form of technology and the expectat ions process, however, t h i s ana l y t i ca l representat ion can be e x p l i c i t l y derived and emp i r i ca l l y implemented. This i s the approach pursued i n Sect ion III.A where the determin is t i c theore t i ca l model of investment dec is ions given a general form of non-stat ic expectat ions formation i s cons t ruc ted . It therefore remains to determine poss ib le forms of the expectat ions formation process. I I .C. The Non-Static Expectat ions Framework " . . . the r a t i o n a l i t y of a person 's choice does not depend upon how much he knows, but only upon how well he reasons from whatever information he has, however incomplete. Our dec is ion i s pe r f e c t l y ra t iona l provided that we face up to our circumstances and do the best we can . " (Rawls) It i s c l ea r that a theory of the expectat ions formation process i s c r u c i a l f o r a f u l l ana l y t i ca l s p e c i f i c a t i o n of the f i r m ' s behavior. To form a foundation to incorporate these expectat ions , i t i s necessary to spec i fy what proper t ies a ' s ens i b l e ' expectat ions process should embody. Two important proper t ies are: i ) the expectat ions formation process must be cons i s t en t , i . e . , i t must not d i f f e r from the r ea l i z ed path sys temat ica l l y and widely , and i i ) i t must be e f f i c i e n t ; i t must use the ava i l ab le information f u l l y and e f f i c i e n t l y . 1 0 These requirements have caused many economists to take the "oppos i te " approach to s t a t i c or "naive" expectat ions and to assume that economic agents are f u l l y " r a t i o n a l " in t he i r dec is ion making in the sense -103-that t he i r expectat ions r e f l e c t the p red ic t ions of the re levant economic theory. The basic idea of the ra t iona l expectat ions hypothesis (hereaf ter REH) i s that economic agents who form expectat ions r a t i o n a l l y der ive p red ic t ions of the future which d i f f e r from actual outcomes only by random and independent e r ro r processes. This assumes an u n r e a l i s t i c degree of information on the part of the economic agents. The concept of ra t iona l expectat ions should, in p r i n c i p l e , be app l i cab le to cases in which there are informational cons t r a i n t s , s ince the basic idea i s how best to assemble a given amount of informat ion with the a id of ava i l ab le economic theory. Thus, the assumption of " r a t i o n a l " expectat ions as formulated in the models to date, i s quest ionable . The r a t i o n a l i t y assumption can be in terpre ted as the assumption that fo res igh t on the average i s a c c u r a t e . 1 1 I f economic agents use equ i l i b r ium values as t he i r fo recas t s , then suppl ies and demands w i l l always be equ i l i b r a t ed at the an t i c ipa ted p r i c e s . However, in the presence o f uncer ta in ty , the future pr i ce (or other var iab le ) must be thought of as a random var i ab le with a p robab i l i t y d i s t r i b u t i o n . The future pr i ce has an expected value and in the ra t iona l expectat ions hypothesis the an t i c ipa ted pr i ce equals the expected p r i c e . Thus, one vers ion of the r a t i o n a l i t y hypothesis i s that "the economic agent knows the pr i ce i s a random var iab le 12 and uses in h is dec is ions the true d i s t r i b u t i o n . " Although th i s "average" i n te rp re ta t i on of the r a t i o n a l i t y idea i s s l i g h t l y l ess r e s t r i c t i v e than the "pure" vers ion that economic agents forecast future p r i ces p e r f e c t l y , there seems to be l i t t l e reason for taking th i s opposite extreme view from the s t a t i c expectat ions assumption. For example, K. Arrow has argued: -104-"It i s true that the ra t iona l expectat ions hypothesis impl ies that the outcomes on future markets are well an t i c ipa ted but i t i s hard to see why t h i s should be t rue . The very concept of the market and c e r t a i n l y many of the arguments in favor of the market system are based on the idea that i t great ly s i m p l i f i e s the informational problems of economic agents, that they have l im i t ed powers of information a c q u i s i t i o n , and that pr i ces are economic summaries of the information from the res t of the wor ld. But in the ra t iona l expectat ions hypothesis , economic agents are required to be super ior s t a t i s t i c i a n s , capable of analyzing the future general equ i l ib r ium of the economy." Th is impl ies the importance of a " l ea rn ing process " , which B. Friedman has emphasized: " . . . what i s t y p i c a l l y missing in ' r a t i ona l expectat ions ' macromodels, i s a c l e a r ou t l i ne of the way in which economic agents der ive the knowledge which they then use to formulate expectat ions meeting th i s requirement. Since Muth's conception of ' r a t i o n a l i t y ' d i f f e r s sharply in t h i s respect from other p l aus ib l e concept ions—Rawls ' , f o r example, which simply requires using opt imal ly whatever information i s a v a i l a b l e , i t i s important to ask how people acquire th i s knowledge of the workings of the economic s y s t e m . " 1 4 With no assumption about th i s learn ing process there i s no explanat ion of why short run d i sequ i l i b r ium would e x i s t , i . e . , why there would be er rors in adjustment of quas i- f ixed inputs . In add i t i on , given that there i s short run d i s equ i l i b r i um , the REH cannot expla in whether or how the equ i l ib r ium value would be a t ta ined . Thus, a learn ing process must be postulated to provide an 15 i n t e rp re t a t i on o f how subject ive data would converge to ob jec t i ve da ta . Simple expectat ions formation such as s t a t i c or ext rapo la t i ve forecasts do not incorporate such a learn ing process ; the f i rm never learns that to have myopic assumptions about pr i ces r e su l t s in e r rors and thus lower p r o f i t s (higher costs ) than i f the future were forecasted more accura te ly . Per fect " r a t i o n a l i t y " models do the oppos i te ; no learn ing i s necessary because the agent knows the s t ructure of the economy. In p rac t i ce nei ther o f these models i s a_ p r i o r i p l a u s i b l e . -105-Recently Benjamin Friedman (1979) has shown that optimal l earn ing by a l e a s t squares procedure "which under f am i l i a r assumptions i s cons i s ten t with ' r a t i o n a l i t y ' i n Rawls' weaker sense" leads to expectat ions formation that are not completely ra t iona l in the sense common to recent l i t e r a t u r e . S p e c i f i c a l l y , they do not exh ib i t the key er ror-or thogonal i ty property inherent to the REH and therefore do not y i e l d the c l a s s i c a l macro r e su l t s assoc iated with the REH. Thus, "under ce r t a i n p laus ib le sets of c i rcumstances, the f am i l i a r ' adapt ive ' expectat ions mechanism may well be a useful approximation to the complex expectat ion generation process assoc iated with such optimal l eas t squares l e a r n i n g . " 1 ^ Therefore , a r e s t r i c t e d r a t i o n a l i t y assumption may be a c l o se r approximation to r e a l i t y than the pure REH, p a r t i c u l a r l y given the short run nature of the re levant models and the l im i t ed amount of time se r i es data a v a i l a b l e . Agents may fo recas t va r iab les on the basis of only t he i r own previous values rather than using the en t i re economic s t ruc tu re , truncated at some po in t in the past . This can be represented by " p a r t i a l " r a t i o n a l i t y , an assumption charac ter ized by neglect of the cross-equation r e s t r i c t i o n s assoc iated with the er ror-or thogona l i t y assumptions. Adaptive expectat ions i s a spec ia l case of t h i s , where the lag i s truncated at the previous pe r iod . This r e s t r i c t e d model may even s a t i s f a c t o r i l y charac ter ize the important observed c h a r a c t e r i s t i c s of the non-stat ic expectat ions formation p rocess . 1 ^ In the context of my model of the f i r m , t h i s impl ies that the i n t e r r e l a t i o n s h i p between stock behavior and expectat ions formation may not be well charac te r ized by the REH; i f expectat ions were t ru l y r a t i o n a l , agents' dec is ions would always be opt ima l , so stock behavior would lose much of i t s importance. An "adapt ive" s t ruc ture of expectat ions on the other hand, impl ies that impacts of shocks in the economy w i l l only be e f f e c t i v e in the short run while people learn and change t he i r expectat ions to r e f l e c t the true -106-s i t u a t i o n ; t h i s adaptat ion of expectat ions over time imposes r e s t r i c t i o n s on the form of the time lag on the adjustment process and thus allows an 18 important ro le fo r learn ing and fo r s tocks , and a l so f a c i l i t a t e s empir ica l implementation. Thus I emphasize here " pa r t i a l r a t i o n a l i t y " assumptions which are conceptual ly extended "adaptive-type" expectat ions formation mechanisms of which the conventional adaptive expectat ions process i s a spec ia l case . -107-III. The Model: Theoret ica l Foundations "Without strong theore t i ca l j u s t i f i c a t i o n for a pa r t i cu l a r form of lag d i s t r i b u t i o n , and perhaps even a strong p r i o r b e l i e f about the quant i ta t i ve proper t ies of that d i s t r i b u t i o n and the fac tors on which those proper t ies depend, i t i s genera l ly impossible to i s o l a t e the lag d i s t r i b u t i o n in any very d e f i n i t i v e way from the sor t of data genera l ly a v a i l a b l e . I t i s t h i s lack of a wel l-def ined imp l i ca t ion of the data that accounts for the s e n s i t i v i t y of the estimates to changes in the sample per iod and var ious assumptions concerning the absence or presence of se r i a l c o r r e l a t i o n among the disturbances of the r e l a t i onsh ips to be estimated that has been found in numerous empir ica l i n ve s t i ga t i ons . The issue i s whether we can in f ac t obtain a bet ter and t ru l y dynamic theory that w i l l determine the form and perhaps even some parameters of the lag d i s t r i b u t i o n s encountered in empir ica l c o n t e x t s . " I 9 These remarks by Marc Nerlove emphasize the need for a mutually support ive r e l a t i onsh ip between theory and empir ic ism. The attempt of the theore t i ca l model spec i f i ed here i s to provide r igorous theore t i ca l underpinnings fo r both a pa r t i a l adjustment process for quas i-f ixed inputs based on costs of adjustment and an e x p l i c i t expectat ions formation process, within an integrated framework that i s emp i r i ca l l y implementable. I pursue each of these components of the f i rm ' s dec is ion process in tu rn , f i r s t the s t ruc tura l model of intertemporal opt imizat ion based on unspec i f i ed expected paths of future exogenous va r i ab l e s , and then the e x p l i c i t cha rac te r i za t ion of the expectat ions formation process. Each of these steps r esu l t s in a form of lag s t ructure for the cha rac te r i za t ion of the f i rm ' s decision-making, based on e x p l i c i t theore t i ca l foundat ions. I I I .A. The Determin is t i c Opt imizat ion Problem with Non-Static Expectat ions The f i r s t step in const ruc t ing the model i s to s t ructure the f i r m ' s de te rmin i s t i c cost minimizat ion problem given ( i ) increas ing marginal costs o f adjustment, and ( i i ) that the en t i r e future paths of exogenous var iab les are incorporated into dec is ions of optimal accumulation of quas i-f ixed inputs . -108-As i n the "s ta te of the a r t " basic dynamic model out l ined in Essay 1, the model formulat ion fo r th i s case i s based on the assumption that the f i rm minimizes costs given expected output, faces in terna l costs of adjustment on net investment, i s a per fec t competitor in input markets, and has a 20 non-constant returns to scale production funct ion fo r i t s output. Thus the f i rm i s assumed to minimize the present value of a stream of future costs at time t : 2.3.1) L(0) = / e ' ^ ^ j P j V j +E 1-a i-z i ) Jo where r i s the f i rm ' s real discount rate (assumed constant fo r a l l t ) , z^ i s the gross add i t ion to the stock of the i t h quasi f ixed input , i . e . , z-(t) = x^ (t) + d . x . ( t - l ) , a.j = u./(r+d.) = the asset pr i ce of x^, u^ i s the rental p r i c e , d . i s the deprec ia t ion ra te , JP.. t + sj^ =Q 1 S t n e known sequence of pr ices of the var iab le inputs v.., and | a . t + sj^=o 1 s the known purchase cos t over time of the quas i-f ixed inputs x^. The f i rm chooses sequences of va r iab le and quas i- f ixed inputs ^vt+s}*s=0 a n c ' { xt+s}s=0 t 0 m 1 ' n 1 " m i " z e c o s t s , given an exogenous known sequence of output demands JYt+sjs=o , the pr i ce sequences over t ime, and x t _ ^ . Internal cos ts in terms of decreased production within the f irm when x i s changed are represented by the presence of x in the production funct ion (Y=f (x ,x ,v , t ) ) such that af/3Jx|<0 and 3 f /s jx j <0. P i s def ined as the vector of va r i ab le input pr ices P.. The two step minimizat ion problem i s s t ructured f i r s t by determining the normalized r e s t r i c t e d cos t funct ion G incorporat ing the so lu t ion to the short run cos t minimizat ion problem, and second, by opt imizing over the quas i-f ixed f a c t o r s . As in the previous Essay, G (P ,x ,x ,Y , t ) i s derived as the normalized va r i ab le cos t funct ion which, because i t i s the so lu t ion to the cost minimizat ion problem, has the p roper t i es : -109-2.3.2) a G / a P . = v . , the optimum short run demand fo r the var iab le input , J J 2.3.3) 3G/3X. = u . , the shadow value of the quas i-f ixed input . The minimizat ion cond i t ions imply cos t l e ss instantaneous adjustment of the var iab le inputs so that r a t i os of the marginal value products equal r a t i o s of current p r i c e s . However, given nonseparable adjustment costs in terms of in terna l costs of adjustment fo r quas i- f ixed inputs (the cap i t a l s tock ) , the leve l o f employment may a f f e c t the costs of investment so that the marginal product of labor w i l l depend on current investment as well as on l eve l s of the va r i ab le and quas i- f ixed fac to r inputs . Hence, s ince investment depends on expected values of exogenous v a r i ab l e s , r e l a t i v e pr i ces and output, both 21 va r i ab le and quas i- f ixed inputs are dynamically determined. The opt imizat ion problem for the quas i- f ixed inputs fac ing the f i r m , given G, i s to f i nd the time path of x ( t ) , k i t ) , which minimizes the present value o f costs over a l l poss ib le G ' s : 2.3.4) min L(0) + £ \ a^x^O) = / e " r t (G(P ,x ,x ,Y , t ) + ux) d t . Th i s problem obviously requi res knowledge of the ent i re future time paths of the exogenous v a r i ab l e s , Y, P, and a, as spec i f i ed in the basic dynamic model i n Essay I. However, in that case s t a t i c expectat ions were assumed, in the sense that Y, P, and a were known to be constant forever at the cur rent va lues , so Y=P=a=0. In the non-stat ic c e r t a i n expectat ions case the time paths of these var iab les do not in general s a t i s f y t h i s equa l i ty . The minimizat ion problem in th i s more general case can be solved e i the r in terms of the non-autonomous maximization p r i n c i p l e or the Euler cond i t i ons . U t i l i z i n g the l a t t e r approach, minimizat ion of (2.3.4) y i e l d s the Euler f i r s t order cond i t i ons : -110-2.3.5) aJ/ax = d/dt(3J/3X), so, 2.3.6) (G +u )e~ r t = d / d t ( e " r t G . ( x , x , P , Y , t ) ) A A = e " r t ( - r G . + G- x + G.-x + G. n P + G- Y + G . . ) , x xx xx xP xy x t 7 ' which co inc ides with the s t a t i c expectat ions so lu t ion from before when P and Y are set to zero . The so lu t ion of th i s d i f f e r e n t i a l equation system i s subs tan t i a l l y more compl icated, however, because now x* = x* ( t ) , where before i t was constant . To see t h i s , wri te the second order d i f f e r e n t i a l equation cha rac te r i z ing the Euler cond i t ion as : 2 ' 3 ' 7 ) G x x k ' + G x x * + G x P P + G x y Y + G x t - r>G. - G x - u = 0, and expand l i n e a r l y around (x ,x , P ,Y , t )= ( (x ( t ) * ,0 , P ( t ) , Y ( t ) , t ) at time t , where a t the expansion point x*(t) i s defined as : 2.3.8a) - r G . * - G * - u = 0, or b) u =-rG<* - G * X X X A This charac te r izes the stat ionary or steady state so lu t ion at time t fo r the equ i l ib r ium stock of x, x* ( t ) , assuming stat ionary expectat ions on exogenous va r i ab les in the steady state (x=x=P=Y=0). Given only one x (=K, the cap i t a l stock) K*(t) i s the "des i r ed " cap i t a l stock at time t as def ined above. Note that given constant P, Y and a (and thus u) , K*(t) corresponds to the standard d e f i n i t i o n of "des i r ed " cap i t a l stock. The i n t u i t i v e i n te rp re ta t i on fo r t h i s cond i t ion i s that the marginal benef i t to the f i rm of changing cap i t a l (the cos t reduction) i s equal to the marginal d i r e c t costs plus the amortized marginal adjustment cos t of a change in the flow of cap i t a l serv ices a t x=0. This adjustment cos t i s considered a cost of changing x and therefore i s added to the r i gh t hand s ide of (2.3.8b) to obtain to ta l user c o s t . -111-The sequence of demands fo r x can be generated from (2.3.7) as an approximate so lu t ion in the neighborhood of x*(t) to a mu l t i va r i a te l i nea r d i f f e r e n t i a l equation system s im i l a r to x= x 2 (x*-x) before , given that x 2 i s the stable root of the quadratic form der ived by a l i n ea r approximation to (2 .3 .7 ) . The l i n ea r expansion of (2.3.7) i s : 2.3.9a) J d x + J Y dx + J p dP + J dY + J . d t + J..dx+ J-dY + J^dP = " ( G x x + r G x x ) ( x ( t ) - x * ( t ) ) + ( G x x - r G x x - G x x ) x ( t ) + G x x x ( t ) + G. p P(t ) + G- yY(t) = 0, or b) G.-x(t) -rG. x(t ) - (G +rG. )x(t ) xx xx xx xx = a 1 x ( t ) + a 2 x ( t ) + a 3 x ( t ) = " ( G x x + r G x x ) x * ( t ) - G x p P ( t ) - G x y Y ( t ) = h ( t ) . Thus the general so lu t ion to the second order d i f f e r e n t i a l equation system i s analogous to previous models, but now i t i s non-homogeneous. Th is impl ies that a " p a r t i c u l a r so lu t i on " ex i s t s that must be dea l t w i th . As in a l l non-homogenous d i f f e r e n t i a l equation cases th i s so lu t ion can be obtained by subs t i tu t ing in any so lu t ion which i s " a " so lu t ion to the equat ion. Hence the so lu t ion to th i s can be der ived a n a l y t i c a l l y by using " jud i c ious guesswork" to 22 solve the in tegra t ion problem. For example, i f (2.3.9) i s rewr i t ten as : 2.3.10) x(t) - rx( t ) - ( ( G x x + r G x x ) / G x x ) x ( t ) = h ( t ) / G x x , the r e su l t i ng d i f f e r e n t i a l equation takes the form: -112-2.3.11) x(t ) + ax(t) + bx(t) = g ( t ) , with a=-r and g ( t )=h ( t ) /G Y Y . In add i t i on , b<0 s ince b i s a product of the roo t s . The roots of t h i s d i f f e r e n t i a l equat ion, x^ >0, X2<0, are so lu t ions 2 to the quadrat ic form x +ax+b = 0. Thus x 2 i s the stable root of the homogeneous d i f f e r e n t i a l equation s im i l a r to the s t a t i c case of the previous Essay. Using " jud i c ious guesswork" as mentioned above, and assuming the unstable root from a t r ansve rsa l i t y cond i t ion that requires l i m t > 0 -rt e G-=0 can be ignored as the system must converge, a so lu t ion of the form A 2.3.12) x(t ) = x 2 x ( t ) + &Jte9{S~t] 9<s)ds can be t r i e d , where B and e are parameters to be determined. This so lu t ion recognizes that i f one has an i n f i n i t e sum to maximize (or minimize) , some assumption must be made to ensure that the sum converges; the negative root must be the one with the dominant impact. This assumption i s equivalent of course to the t r ansve r sa l i t y cond i t i on . D i f f e r e n t i a t i n g (2.3.12) one obta ins : 2.3 13) x(t ) = x 2 x ( t ) - Be(t) - Be jtee^S~l) g(s) ds . Subst i tu t ing from (2.3.12) y i e l d s : /oo r oo t e e ( s _ t ) g(s)ds - Be(t) - eB L e e ( s _ t ) g(s)ds (2.3.12) and (2.3.14) can then be subst i tu ted in to (2.3.11) to obta in e and B by comparing c o e f f i c i e n t s : t e e ( s _ t ) g ( s ) d s - Be(t) + ax 2 x ( t ) + as / t e e ( s ' t ) g ( s ) d s + bx(t) -113-=(X 0B - 66 + ae) e(s-t) g(s)ds - ee(t) = g ( t ) . Th i s u t i l i z e s the f a c t that (x 2+ax 2+b)x(t)=0. Comparing c o e f f i c i e n t s , I ob ta in : B=-1, X/,-e+a=0, so e=xo+a<0 s ince a=-r and x ? <0. Hence the so lu t ion becomes: 2.3.16a) kit) = x 2 x ( t ) -( x 9 - r ) ( s - t ) e c g(s)ds , o r , b) kit) = x 0 x( t ) -J x 9 - r ) ( s - t ) e 2 (h(s)/G. . )ds , in the o r i g i na l no ta t ion . Thus the stable so lu t ion path for investment i s obtained as an "adapted" f l e x i b l e acce le ra tor form. Th is expression can be r e spec i f i ed s l i g h t l y to der ive a form more c l o se l y analogous to (2 .1 .2 ) . Given the d e f i n i t i o n of h(s) ( restated here) : 2.3.17) h(s) = - (G x x +rG. x ) x * ( s ) - G x p P ( s ) - G . y Y ( s ) , and subs t i tu t ing x fo r x 2 , (2.3.16b) can be expanded to become 2.3.18) kit) = xx(t) + (1/G..) t e ( X " r ) ( S " t ) ( ( G x x + r G x x ) x * ( s ) + G . p P ( s ) + G x y Y ( s ) ) ds . Consider the in tegra l over x* ( s ) , which can be restated in terms of the i n f i n i t e sum of x*(t) c a l cu l a t ed in per iod t given constant expectat ions on a l l future exogenous va r i ab l e s , plus the i n f i n i t e sum of va r i a t ions in x*(s) from x*(t) fo r a l l future time periods s, i . e . , 2.3.19) x*(s) ds = x*(t) ds + Subst i tu t ing t h i s into (2.3.18) r e su l t s i n : -114-2.3.20) x(t ) = xx(t) + (1/G-) 7 + e U " r ) ( s " t ) ( G x x + r G x x ) x * ( t ) + ( 1 / G x x > / t € + { 1 / G x x } / ^ ( X ' r ) ( S ' t ) ( G x x + r G x x ) ( x * ( s ) - x * { t ) ) + G x p P ( s ) + G x y Y ( s ) ds . x*(t) i s constant , however, so the second expression on the r igh t hand side of (2.3.20) can be s i m p l i f i e d s u b s t a n t i a l l y , and then rewri t ten as : 2.3.21) ( l / a ^ / t e { x " r ) ( s _ t ) a 3 x* ( t ) ds = - a ^ M t J / a ^ x - r ) = -bx*(t)/(x-r) = (-bx*(t) x )/ (x 2 -rx) = (-bx*(t) x)/-b (given x 2 + ax + b = 0 and a=-r) = xx ( t ) * . Thus (2.3.20) becomes: 2.3.22) kit) = xx(t) - xx*(t) + (1/G-.) f t e ( x " r ) ( s " t ) ( ( G x x + r G x x ) ( x * ( s ) - x * ( t ) ) + G x p P ( s ) + G x y Y ( s ) ) d s « = |x| (x*(t) - x(t ) ) + (1/G..)y" te ( x" r ) ( s" t )((G x x+rG. x)(x*(s)-x*(t)) + G- P(s) + G* Y(s)) ds , = /x| (x*(t) - x(t ) + J) = I x \ (x**(t) - x ( t ) ) , s ince x<0. J spec i f i e s e x p l i c i t l y what port ion of investment demand corresponds to non-stat ic expecta t ions . Note that with s t a t i c expectat ions -115-th i s f i n a l term J drops out , as x*(s)=x*(t)=x* and Y=P=0, leav ing only the s t a t i c expectat ions form of the cap i t a l accumulation equat ion. Given x as a system of parameters determined by the negative roots of the quadrat ic form (or one parameter in the one quas i-f ixed-input case ) , the so lu t ion der ived here, as in the analogous d i sc re te N i cke l ! (1977) 23 formulat ion , ind ica tes that with non-static expectat ions : " . . . i n s t e a d of aiming at a s ing le equ i l ib r ium leve l of cap i t a l s tock, the f i rm aims at the equ i l ib r ium cap i t a l stock based on current p r i ces plus an exponential weighted sum of the d i f fe rences between th i s equ i l ib r ium cap i t a l stock and the d i f f e r e n t equ i l ib r ium cap i t a l stocks corresponding to the l e ve l s of demand and pr i ces expected fo r a l l future pe r iods " . Note that an e x p l i c i t s t ructure on the lag formation has been generated by the unobservable intertemporal dec is ions about the f i rm ' s "des i red " stock of c a p i t a l . This s t ructure depends on the expected future paths of the exogenous va r i ab l e s . I I I.B. A l t e rna t i ve Ana l y t i ca l Spec i f i c a t i ons fo r Expectations Formation The procedures discussed above speci fy the st ructure of the de te rmin is t i c behavior of the f i rm over time with a changing target va r iab le given ce r t a i n expectat ions of future exogenous va r i ab l e s . I now turn to a d iscuss ion of the second s tep, the cha rac te r i za t i on of the expectat ions formation process. Without some extreme s i m p l i f i c a t i o n s on the s t ruc tu re , true " r a t i o n a l " expectat ions models requ i r ing exact ce r t a in t y equivalence are too complicated 24 to impose on the s t ruc tura l model above. Such r e s t r i c t i o n s are a p r i o r i unacceptable given that my emphasis i s on the s t ruc tura l economic model. As argued above, however, simpler forms of the expectat ions formation process can provide a useful approximation to ra t iona l expectat ions. The arguments used a n a l y t i c a l l y to j u s t i f y t h i s approach are based on two "approximations" to r a t i o n a l i t y . The most important i s the re laxa t ion of -116-exact ce r ta in ty equivalence requ i r ing a l inear-quadrat ic dec is ion c r i t e r i o n funct ion and l i nea r cons t r a i n t s . The second approximation i s the assumption that exogenous var iab les are pred ic ted only on the i r own past values rather than on other var iab les in the system; t h i s has been c a l l e d " pa r t i a l r a t i o n a l i t y " . Given that these var iab les are exogenous to the decision-making process, that there i s no s imultanei ty problem in the model, and that the expected values of these exogenous va r i ab les depend only on the i r own past va lues , the usual r e s t r i c t i o n s impl ied in more complicated simultaneous equation systems from macro models can be ignored. This has been c a l l e d "quas i - r a t i ona l " expectat ions . Further r e s t r i c t i o n s r e su l t in "adaptive expec ta t ions " , which have a lso been suggested as a useful approximation to ra t iona l expectat ions . Even simpler spec i f i c a t i ons can be der ived from imposing r e s t r i c t i o n s on the adaptive expectat ions model. The d i v i s i o n of the opt imizat ion problem into two components, ( i ) the opt imizat ion of the object ive funct ion given expectat ions, and ( i i ) the determination of expectat ions format ion, r e l i e s on ce r ta in ty equiva lence. In order to have the ce r t a in t y expectat ions assumptions s a t i s f i e d , s t r ingent s imp l i f y ing cond i t ions on the form of the object ive funct ion must apply . Nerlove (1972) s p e c i f i e s the bas is fo r t h i s problem: "Unfor tunate ly , unless the production funct ion s a t i s f i e s very s t r ingent cond i t i ons , we cannot, i n genera l , solve t h i s problem by rep lac ing the s tochast i c va r i ab les by t he i r cond i t iona l expectat ions. Th is i s because such a s t ra tegy , even i f adopted only fo r the f i r s t , instantaneous future pe r i od , does not al low the opt imizer to take f u l l advantage of the f a c t that at subsequent points in time he willj-know some of the values of the s tochast i c va r i ab les present ly unknown." Although true " ce r ta in t y equivalence" cannot be j u s t i f i e d in my model, i t may be useful to assume tha t , at l e as t f o r each ind iv idua l pe r i od , dec is ions made by opt imiz ing the object ive funct ion with the s tochast ic va r iab les replaced by t he i r cond i t iona l expectat ions are approximately those which would r e s u l t i f the expected values were maximized. Nerlove points out : -117-"Malinvalid (1969) shows that th i s i s the case , under c e r t a i n d i f f e r e n t i a b i l i t y cond i t ions on the object ive func t ion , provided the s tochast i c elements of the problem are def ined in such a way as to have zero expectat ion and are not ' too l a r g e ' . This may be j u s t i f i e d by d iscount ing at a s u f f i c i e n t l y high rate to ensure that the d i s t an t fu tu re , where uncerta inty i s the greates t , w i l l matter very l i t t l e . A l t e r n a t i v e l y , one might simply argue that a model der ived from maximization of (the object ive funct ion) with the values of s tochast i c va r i ab les replaced by t he i r cond i t iona l expectat ions i s l i k e l y to prove great ly super ior from an empir ica l p o i n t g o f view to any model based on the assumption of s t a t i c expec ta t ions . " Thus, even i f the ce r ta in t y equivalence assumption does not s t r i c t l y ho ld , i t may charac te r ize r e a l i t y f a i r l y we l l . It may be a " c lose enough" approximation to use rather than to impose an over ly s i m p l i s t i c s t ructure on the model to ensure that the cond i t ions for ce r t a in t y equivalence hold 27 exac t l y . Based on such reasoning I assume in th i s Essay that e i t he r expectat ions are held with ce r t a in t y or that an "approximate" ce r t a in t y equivalence can be assumed so that I take in to account only the determin is t i c part (expected value) o f the expectat ions formation process. In order to determine what types of ana l y t i ca l expectat ions formation processes might be imposed, i t i s useful to consider how the var ious a l t e rna t i ve assumptions a f f e c t the model. E x p l i c i t incorporat ion o f expectat ions formation in dynamic models has, as mentioned above, resu l ted in models based on d i s t r i bu ted l ags , o f t en , as i n the case of recent ra t iona l expectat ions models, becoming extremely complex. In simple s p e c i f i c a t i o n s , these d i s t r i bu ted lag formulations are based on the model: 2.3.23) y t = a + b x * t + u t , where x*^ could be the expected value in the adaptive expectat ions model. Nerlove (1972) (and others) show that even with more complicated expectat ions cha rac te r i za t ions t h i s cha rac te r i za t ion usua l ly can be reduced to a ra t iona l 28 d i s t r i bu t ed lag : -118-2.3.24) y t = a + b(N(•)/D(•))x t + u t , where N(-) and D(-) are polynomials in the lag operator and x t i s an observable exogenous va r i ab l e . The f ac t that the lag d i s t r i b u t i o n i s ra t iona l al lows the s t ruc tura l equation to be transformed in to one conta in ing only a 29 f i n i t e number of v a r i ab l e s , thereby f a c i l i t a t i n g es t imat ion . S p e c i f i c a l l y , i t can be shown that " r a t i ona l d i s t r i bu ted lags a r i s e when the exogenous var iab le (or i t s pth d i f fe rence ) has a mixed autoregressive-moving average representat ion and x * t i s chosen to minimize the expected value of a quadrat ic ob jec t i ve func t ion . For example, x * t could be the l e a s t squares 30 fo recas t of x t made at time t-v." Thus, the expression of expected var iab les in terms of a ra t iona l d i s t r i bu t ed lag of past var iab les can be charac ter ized by time ser ies ana lys i s techniques, i . e . , by an ARIMA model. The subs t i tu t ion of x t * in to the simple s t ruc tura l model (2.3.23) i s analogous to the procedure required fo r my model based on the subs t i tu t i on of some funct ion of past observed values Y t _ k and P t _ k f o r the expected based on the en t i r e future paths of the exogenous va r i ab l e s , however, th i s i s more complex than in the case charac ter ized by (2 .3 .21) . The f i r s t step toward a t r ac tab le economic model i s to formulate conceptual ly how an t i c i pa t i ons at any one time are formed by economic agents. In the recent l i t e r a t u r e on ra t iona l expectat ions, models of an t i c i pa t i ons are generated as cond i t iona l expectat ions using time se r i es ana lys i s techniques, s im i l a r to the general process represented in (2.3.21) by the ra t iona l d i s t r i bu t ed lag on x ( t ) . A l t e r n a t i v e l y , var ious simpler hypotheses on expectat ions formation may be imposed. As suggested above, these simple models may be j u s t i f i a b l e ; although REH i s in some sense less r e s t r i c t i v e than 31 simpler models of expectat ions, i t i s a lso more r e s t r i c t i v e s ince i t values Since the intertemporal model i s -119-impl ies behavior cons is tent with per fec t knowledge of the economic model, rather than r e f l e c t i n g a learn ing process. Thus, to c l a r i f y the a l t e rna t i ve approaches and the i r underly ing r a t i o n a l i z a t i o n s , I now d igress b r i e f l y to provide an overview of var ious poss ib le expectat ion formations models proceeding from the s implest models to the "op t ima l i t y " or " r a t i o n a l " models, a l l based within the context of time ser ies fo recas t ing techniques. Further d iscuss ion of Time Ser ies Ana lys is (TSA) techniques can be found in Appendix A. The s implest assumption about expectat ions i s that of s t a t i c expectat ions . S t a t i c expectat ions are adequate i f the f i rm can adjust everything instantaneously so that i t s myopic behavior does not impose any cons t ra in t s on optimal decision-making. If there are any cons t ra in ts on the f i r m ' s opt imizat ion behavior such as de l i ve ry lags or costs of adjustment, however, the assumption of s tat ionary exogenous var iab les over time imposes severe a p r i o r i s t ructure on the f i r m ; i t impl ies the f i rm i s always lagging behind in adjustment towards optimum l e v e l s . This model i s thus useful only under very r e s t r i c t i v e condi t ions as i t impl ies that producers su f fe r constant losses and that they never learn from experience. The only time that i t i s reasonable to assume expected future values of the exogenous var iab les are a l l equal to the observed current values i s where pr i ces are a random walk so that fo recas t ing information from other past values i s nonexistent , a cond i t ion which in th i s context i s not very p l a u s i b l e . The adaptive expectat ions mechanism was proposed by Cagan in 1956. According to th i s hypothesis , economic agents are assumed to rev ise t he i r expectat ions according to recent exper ience: 2.3.25) P* t - P* t_ x = yiP^ - P * ^ ) , where P*+_i i s the expected p r i ce fo r per iod t-1 at time per iod t-2 and y i s -120-the c o e f f i c i e n t of expectat ions . S ta t i c expectat ions are a lso a specia l case of t h i s model when y=l so p * t = p t _ i « C o l l e c t i n g terms, rep lac ing (1-y) by B , and assuming B<1, one can der ive a more common expression fo r adaptive expectat ions as : 2.3.26) P* t = ( l - B ) . E r = 0 B k p t - l - k -The adaptive expectat ions models can therefore y i e l d d i s t r i bu ted lag models, for with adaptive expectat ions the expected pr i ce can be represented by an i n f i n i t e weighted average of past r e a l i z ed p r i ces with geometr ica l ly dec l i n ing weights. Th is model r e su l t s in a rather r e s t r i c t i v e s p e c i f i c a t i o n for the weights on the lag are constra ined to dec l ine geometr i ca l l y . It i s therefore b a s i c a l l y ad hoc, as there appears to be no theore t i ca l j u s t i f i c a t i o n f o r a 32 geometr ica l ly dec l i n ing lag s t ruc tu re . The adaptive expectat ions hypothesis , however, can be viewed as a specia l case of the " r a t i o n a l " d i s t r i bu t ed l a g , which can be in terpre ted as a cha rac te r i za t i on of REH. Th is can be s p e c i f i e d a n a l y t i c a l l y using TSA methods. I f ra t iona l expectat ions can be generated by a general ARIMA model and i f adaptive expectat ions in equation (2.3.26) can be charac ter ized by an IMA (1,1) model as discussed in Appendix A, then adaptive expectat ions i s a "nested" cha rac te r i za t ion with in the general ARIMA framework. The geometric lag form i s therefore a r e s t r i c t e d form of the general " r a t i o n a l " lag that could charac te r ize the d i s t r i bu t ed lag process. Rational expectat ions hypotheses evolved from th i s type of ana l y s i s , where ra t iona l expectat ions are formed as the cond i t iona l expectat ions of the var iab le being fo recas t based on a l l past observat ions on i t and i t s re la ted va r i ab l e s . Th is i s the sense in which the hypothesis assumes substant ia l s t a t i s t i c a l knowledge of the s t ructure generating the observed data . For -121-sta t ionary processes, minimum mean-square-error forecasts and cond i t iona l expectat ions are equ iva lent , so a cha rac te r i za t ion of the s tochast i c s t ruc ture can be generated f a i r l y e a s i l y , p a r t i c u l a r l y when the forecasted var iab les are exogenous. A n a l y t i c a l l y the REH impl ies that the expected pr i ce fo r per iod t+1 at time t i s equal to the expected value ( in a s t a t i s t i c a l sense) of the equ i l ib r ium value of the p r i ce in per iod t+1, cond i t iona l on information up to per iod t , i . e . , 2.3.27) P * t + i = E t ( P t + 1 ) . Thus expectat ions are unbiased and the expected pr i ce i s endogenous to the system. This assumption can be incorporated into the model by appending (2.3.27) to the system of equations cha rac te r i z ing the economic agent 's dec i s ion formation process. In a sense, the endogeneity of t h i s expected path i s what complicates the ra t iona l expectat ions hypothesis . Endogeneity of the expected pr i ce r esu l t s i f the cur rent p r i ce i s endogenous to the model, as i t i s in most macro or general equ i l ib r ium supply and demand systems. This generates the usual REH cross-equation r e s t r i c t i o n s , implying that the c o e f f i c i e n t of expectat ions depends on the s t ruc tura l parameters of the system. However, i f the actual p r i ce leve l can be asssumed to be exogenous, i t i s reasonable to assume that the expected pr i ce can be charac te r ized in terms of only past values of th i s va r i ab le rather than depending on other va r iab les of the system. Thus the expected pr i ce leve l determined by the economic agent i s exogenous to the system; i t depends only on exogenous va r i ab l e s . In other words, although the agent forms expectat ions about the s tochast ic s t ructure and therefore the time path of the expected v a r i ab l e s , t h i s process can be separated from the formation of optimal dec is ions over time since these dec is ions w i l l not a f f e c t -122-the expected p r i c e . Th is has been c a l l e d " pa r t i a l r a t i o n a l i t y " in the l i t e r a t u r e , s ince i t depends only on the past values of the var iab le i t s e l f rather than on cross-equation r e s t r i c t i o n s . In essence, true ra t iona l expectat ions are t heo re t i c a l l y j u s t i f i a b l e s ince they can be formulated as the outcome of an opt imizat ion process and therefore in some sense charac te r ize an "opt imal " expectat ions formation process. However, in other senses as I have d iscussed , the REH i s ac tua l l y more r e s t r i c t i v e . It i s therefore not obvious that the assumption of " t rue r a t i o n a l i t y " in a model i s the most useful approach, t h e o r e t i c a l l y £ r in terms of app l i c a t i on . Thus, I consider " p a r t i a l " ra t iona l expectat ions and adaptive expectat ions to be reasonable and v iab le a l t e rna t i ve spec i f i c a t i ons and emphasize these cases , although given the assumptions in my models the " p a r t i a l " versus " t rue " r a t i o n a l i t y a l t e rna t i ves are not c l e a r l y 33 d i s t i ngu i shab l e . Given the assumptions of approximate ce r ta in t y equivalence and a time ser ies s t ructure of past values of the own var iab le fo r expected exogenous va r i ab l e s , the model can be implemented. The basic procedure i s to obtain the e x p l i c i t form of the an t i c ipa ted var iab les as funct ions of the past exogenous var iab les using time se r i es techniques. In the adaptive expectat ions case the an t i c ipa ted se r i es may then be obtained by const ruc t ing the cond i t iona l expectat ions for the exogenous va r i ab l e s . These ser ies can then i m p l i c i t l y be incorporated in to the general model by subs t i tu t ing the re levant parameters and " innovat ions" from the process in to the s t ructura l demand equations represent ing the f i r m ' s behavior. In the more general expectat ions case , where the " t rue" ARIMA processes generating the var iab le are unknown, analogous procedures can be app l ied by determining the order of the ARIMA process and c a l c u l a t i n g the impl ied cond i t iona l expectat ion. A l t e r n a t i v e l y , an unspec i f i ed ARIMA form can be approximated by a l lowing a compound parameter -123-depending on e r rors in expectat ions in to the system of demand equations to r e f l e c t the to ta l e f f e c t of expectat ions . Although the s t ruc tura l expectat ions parameters cannot be i d e n t i f i e d in th i s case , the r e su l t s are cons i s ten t with the " p a r t i a l r a t i o n a l i t y " assumption, which I c a l l " r a t i o n a l " . I emphasize the l a t t e r approach, which I c a l l the "genera l " model, and the adaptive expectat ions formulat ion as a specia l case . The ana l y t i c s involved in these s p e c i f i c a t i o n s w i l l be c l a r i f i e d in Sect ion IV. I I I .C. Empir ica l Spec i f i c a t i on In order emp i r i ca l l y to implement the model discussed in the previous two sec t i ons , and s p e c i f i c a l l y to incorporate adaptive or " r a t i o n a l " expectat ions in to th i s model, one addi t iona l approximation beyond those discussed in Essay 1 and those discussed in the previous sect ion must be imposed on the s t ruc tu re . The f i n a l step i s to transform the s t ruc tura l model with an i n f i n i t e continuous lead to a t rac tab le form. This requires approximating the continuous model by a d i s c re t e form, and t runcat ion e i the r of the decision-making p r o f i l e or transformation to decrease the number o f re levant future and past parameters. F i r s t , keeping in mind the problems involved with aggregating over time which are discussed in Essay 1, I a l t e r the continuous expression fo r (2.3.22) with one quas i-f ixed input , cap i t a l (K): 2.3.28) K(t) = xK t_j_ - \K* t - j t e ( x - r ) ( s _ t ) ( h ' ( s ) / G ^ ) d s , where h 1 (s)=-((G K K+rGfc K ) (K*(s)-K*(t))+G$ P(s) + Gj^Y(s) ) , to a d i s c re te form to correspond to data requirements. The problem a r i s e s due to the in tegra l on the r i gh t hand s ide . The s implest method of approximating t h i s in a d i s c re t e form i s to a l t e r the in tegra l form to read: -124-2.3.29) X)^ IT l / ( l - ( x - r ) ) s " t + 1 ) ( h ' ( s ) / G ^ ) . Thus, approximating K by K t ~ K t _ ^ or A K ^ . , one obta ins : 2.3.30) AK(t) = x K t _ 1 - \K* t - £ ~ t l / ( ( l - x + r ) s ' t + 1 ) h ' ( s ) / G ^ . This remains in the form of an i n f i n i t e sum however which, s i m i l a r l y to i n f i n i t e d i s t r i bu t ed lag s e r i e s , cannot be estimated due to the absence of an i n f i n i t e number of degrees of freedom. Three a l t e rna t i ve methods can be used to deal with th i s problem. One i s to truncate the lead a r b i t r a r i l y , assuming that the re levant time horizon fo r the decision-making process i s f i n i t e and minimal. The second i s to assume that the i n f i n i t e sum converges to a f i n i t e expression and then u t i l i z e t h i s l i m i t i n g value to summarize a l l the information (the 'convergence' procedure) . The t h i r d i s to use a procedure analogous to a Koyck transform for d i s t r i bu t ed lags and apply i t to the d i s t r i bu t ed lead formation (the Koyck 34 procedure) . I now discuss each of these procedures in tu rn , and use the t h i r d fo r the empir ica l s p e c i f i c a t i o n . There are various ind i ca t ions that the f i rm w i l l use a l im i t ed time hor izon to make i t s current dec i s i ons . One i s that even with rap id ly increas ing costs of adjustment, the f i rm w i l l not spread i t s adjustment of quas i- f ixed inputs over a very long per iod of t ime, so that the re levant time horizon fo r the f i rm w i l l be the per iod of "most" of the adjustment. In a d d i t i o n , p a r t i c u l a r l y with a high discount r a t e , future dec is ions w i l l be discounted very r a p i d l y . Uncertainty about future expected values of exogenous var iab les w i l l a lso tend to minimize the time horizon of the decision-making process. The f i rm w i l l be inc reas ing ly uncertain about future expected pr ices given current known values as the expectat ions horizon extends in to the fu tu re . Thus i t w i l l l i k e l y not p red i c t large changes in values fa r -125-in the fu tu re . Thus r e l a t i v e l y " s t a t i c " expectat ions of exogenous var iab les w i l l be evident a f t e r a c e r t a i n future date, i nd i ca t ing that a f t e r that point the trends or s t ructure of the disturbances are not as ce r t a i n and can be assumed to be random. The in te rac t i on of the e f f e c t s that cause future dec is ions to be both r e l a t i v e l y unimportant and uncertain ind ica te that t h i s sum could be truncated at some po in t . In p r i n c i p l e i t would be poss ib le to t e s t f o r t h i s cu t-o f f po in t , but given the complexity of the form of the equat ion, and the p o s s i b i l i t y with d i s c re te approximation of an erroneous i nd i ca t i on of long lags (or leads in t h i s case ) , t es t ing for t h i s would be d i f f i c u l t and could be mis lead ing . Thus i t seems more reasonable to truncate the lead somewhat a r b i t r a r i l y , using guidance from both i n t u i t i o n and the values of the discount r a t e , the parameters of the expectat ions process and the empir ica l evidence on the s i ze of the adjustment c o e f f i c i e n t s on the quas i- f ixed inpu ts . The problem with t h i s procedure i s that i t does not ea s i l y s imp l i f y to an emp i r i ca l l y implementable form, even with t runcat ion a t , say, f i v e yea rs . The second p o s s i b i l i t y can be pursued by cons ider ing convergence of the form (2 .3 .29 ) . By const ruct ion ( l / ( l - x + r ) ) < l so that l / ( l - x + r ) s ~ t + 1 has a l i m i t i n g value of zero as s approaches i n f i n i t y . Thus the i n f i n i t e sum of these c o e f f i c i e n t s converges to the l i m i t i n g form l/ ( r-x ) i f mu l t i p l i ed by a constant . However, i t i s not c l e a r what h'(s) converges to or even i f i t does converge in genera l . If pr ices and output, on which K*(s) depends, continue to increase forever then convergence w i l l not be achieved; a l i m i t i n g value w i l l not e x i s t and the i n f i n i t e sum w i l l not be express ib le in a simple form. If a s p e c i f i c form i s imposed on the expected exogenous var iab les that determine h(s) and on the technology represented by G, the cha rac te r i za t ion of t h i s i n f i n i t e sum can be determined given simple converging expectat ions hypotheses. Th is i s poss ib le given ce r t a in t y or ce r t a in t y equivalence with -126-both ext rapo la t i ve and adaptive s p e c i f i c a t i o n s , as well as with s t a t i c expectat ions , the l a t t e r being a t r i v i a l case . In these cases , s ince the convergence of the expected ser ies can be der ived a n a l y t i c a l l y , the r e su l t i ng expression can be spec i f i ed e x p l i c i t l y and subst i tu ted in to the s t ruc tura l model; t h i s i s the "convergence precedure" . Here the ce r ta in t y equivalence assumption i s equiva lent to an assumption that the e r ro r terms on the expectat ions process are independent of the e r ror form spec i f i ed fo r the investment demand process, so that only the determin is t i c part must be s p e c i f i e d and subs t i t u t ed ; the e r ro r on the equation simply becomes a composite of 'white no ise ' e r ro r s . I do not pursue these models in t h i s Essay. However a d iscuss ion of the impl ied est imat ion equations i s reported i n Appendix B. The f i n a l case i s that of a Koyck transform on the d i s t r i bu t ed l ead , s im i l a r to that of Abel (1978). This i s an i n t e res t i ng case which i s d i s t i n c t i n important ways from the d i s t r i bu ted lag formulat ion, s p e c i f i c a l l y s ince the va r i ab les involved in the d i s t r i bu ted lead are unobservable. Thus est imat ion problems e x i s t which depend on the change in expectat ions of future var iab les from new information revealed in the lead time per iod . This method i s the approach I use in the empir ica l implementation. However i t i s more eas i l y d iscussed i n the context of a s p e c i f i c funct iona l form. Therefore fu r ther cons idera t ion of t h i s case i s deferred to the next sect ion a f te r the e x p l i c i t funct iona l forms fo r the f i r m ' s decision-making process are s p e c i f i e d . Thus, given the s imp l i f y ing assumptions and approximations from th i s sect ion and those discussed in Essay 1, a model can be derived that i s emp i r i ca l l y implementable and y e t , hope fu l l y , takes in to cons iderat ion the most important components of both the decision-making and expectat ions formation s t ructure of the f i rm with minimal m i s s p e c i f i c a t i o n . The next sec t ion discusses econometric implementation in the l i g h t of the q u a l i f i c a t i o n s in the model s p e c i f i c a t i o n presented here. -127-IV. Econometric Implementation Assuming there are four inputs , quas i- f ixed cap i t a l (K), and var iab le inputs labor (L ) , energy (E) , and mater ia ls (M), that output Y^ in per iod t i s a funct ion of c ap i t a l in place at the beginning of the pe r iod , K ^ , that a l l input pr i ces are normalized by the pr i ce of labor P|_, and that a l l assumptions in the previous sect ion ho ld , I spec i fy a funct ion G as a quadratic approximation to the normalized r e s t r i c t e d cos t func t ion : 2.4.1) G T = L + P E t E + P M t M = (a Q + a o t t + a E P E T + + a Y Y t + ( 1 / 2 ) ( Y E E P E \ + Y MM P Mt + Y EM P E t P Mt + Y E Y P E t Y t + Y MY P Mt Y t + a £ t P E t t + a M t P M t t ) Y t + aKKt_x+ + ^ ( Y ^ K 2 ^ + Y ^ ( A K T ) 2 ) + Y E K P ^ K ^ ^ Y ^ P ^ K ^ + Y Y K Y t K t - l + Y E K P E t A K t + Y M K P M t A K t + Y Y k V K t + Y K K K t - l A K t + a K t K t - l + a t K t A K f Internal costs of adjustment are represented by: 2.4.2) CUK) = a £ A K T + ( l / 2 ) T ^ ( A K T ) 2 + Y E | < P E T A K T + Y ^ ^ t + T ^ A K ^ + Y K K A K t K t - l + a t K A K t t " Marginal adjustment costs at a s tat ionary state A K t = 0 are assumed to be l i m ^ ^ Q C ' ( A K t ) = 0 . These costs are given by: 2.4.3) G a K = C ' ( A K T ) = a£ + % A K T + y ^ + y t f ^ + ^ + y ^ K ^ + c ^ t , which, at AK=0, r equ i r es : 2.4.4) C' (0) = ^ = Y E K = Y M K = Y Y K = Y K K = ^ = 0. -128-In previous work these r e s t r i c t i o n s have been imposed on a l l funct iona l forms because the f i rm i s viewed as pursuing a given stat ionary K* def ined by AK=0. However, in my formulat ion th i s does not hold because the f i rm i s not pursuing a given K* but a moving target K**(t) charac te r ized by h ( t ) . K**(t) c l e a r l y inc ludes a K*(t) term, however, def ined at the stat ionary point AK=0, so fo r purposes of determining K*(t) the r e s t r i c t i o n s (2.4.4) must s t i l l be imposed. Short run cond i t iona l demand funct ions fo r va r i ab le inputs are obtained using the re levant vers ion of Shephard's lemma and the d e f i n i t i o n of G: 2.4.5) E = 3 G / 9 P E t = ( « E + y E E P E t + y E M P M t + Y E Y Y T + a E t t ) Y t + Y ^ K ^ + Y E f c A K t > 2.4.6) M = 8 G/3P M t = ( a M + Y M M P M t + Y E M P E t + Y MY Y t + a M t t ) Y t + Y MK K t- l + YM|<AK T, 2.4.7) L = G - P E t E t - P M t M t - UQ + a ^ t - < l / 2 ) ( T E E P f t + ^^EM^Et^Mt 2 2 + YMM PMt } + a Y Y t ) Y t + a K K t - l + a K A K t + ( 1 / 2 ) ( Y K K K t - l + Y ^ ( A K t ) 2 ) + Y Y K Y t K t _ 1 + Y Y K Y T A K T + Y KK^t- l A ^t + " K ^ t - l * + a ^ t A K t . In cont ras t to previous work, AK appears in a l l the short run demand equations instead of j u s t in the labor equat ion. However, t h i s does not cause s imultanei ty problems because the Jacobian remains t r i angu la r . Also note that the s p e c i f i c a t i o n of the f i r m ' s technology by a normalized var iab le cos t funct ion causes asymmetry of the input demand equations, s ince the pr i ces are 35 a l l normalized by the pr ice of one a r b i t r a r i l y named input . Given the -129-current s p e c i f i c a t i o n , L. J . Lau (1976) and E . Kokkelenberg (1979) suggest that to minimize the m isspec i f i c a t i on in the stochast ic spec i f i c a t i on from th i s asymmetry i n the determin is t i c model, i t i s useful to normalize by the pr i ce of the most var iab le input . Since L i s often considered to be r e l a t i v e l y f ixed i t may poss ib ly be more useful to normalize by another input p r i c e , say P £ t . However, because previous work has been based on normal izat ion by I r e ta in t h i s assumption. F i n a l l y , the sequence of investment demands must be cha rac te r i zed . The form of the investment equation can be derived as equation (2 .3 .33) , reproduced here with h'(s) subs t i tu ted : 2.4.8) A K ( t ) = x K t - 1 - xK* t + (1/G^) X)S=T ( l / ( l -x+r ) ( s " t ) ) ( ( (G K K +rG | ^ K ) ) ( K * s -K* t ) + G E K P E S + G M K P M S + G Y K V -K * ( s ) i s obtained from the stat ionary so lu t ion (2.3.8)) with s t a t i c expectat ions of a l l exogenous var iab les a t time s. Given the funct ional form for G and the r e s t r i c t i o n s fo r the stat ionary so lu t ion from (2 .4 .4 ) , K*(s) becomes: 2.4.9) K*(s) = ( - 1 / Y K K H « K + Y E K P E S + Y M K P M S + a K t S + Y Y K Y s + U K s K K*^, P.j. and Y t are known at time t , and the expected values of K* s, P. , and Y s depend on the expected occurrences of P.. , Y , and u^ s , which are determined by the spec i f i c funct ional form of the assumed process of expectat ions formation. Using the approximation to ce r ta in t y equivalence assumption, I consider these expectat ions processes independently from the rest of the dec is ion making process in order to separate the f i r m ' s decision-making procedures into two components. -130-As d iscussed above, the cha rac te r i za t ion of the f i rm ' s expectat ions can be formulated in a l t e rna t i ve ways. I employ the "Koyck adjustment" to formulate and estimate the model. In add i t i on , the assumption that the expected value ( in the s t a t i s t i c a l sense) of the disturbance term for the expected var iab les formation i s equal to zero i s required for some sort of " r a t i o n a l i t y " to e x i s t . Everything re levant i s taken into cons iderat ion in the expectat ions formation process s ince only a "white noise" e r ror remains. App l i c a t i on of t h i s approach f i r s t requires expanding equation (2 .4 .8 ) , given the funct ional form G and the expression for h'(s) to ob ta in : 2.4.10) AK t = K t - K t l = xK t_j_ - xK* t - ] £ ~ t ( l / ( l - x W S ~ t + 1 ^ h ' ( s ) / Y | ^ = x K ^ - AK* t + ( l A t f ) E r = t { 1 / { 1 - X + r ) ( S " t + 1 ) ) ( ( Y K K + r % ) ( K * s - K + Y E K A P E s + Y MK A P Ms + M A Y s * where A P ^ P J - P ^ ) , and aY*=Y*-Y*_ r The t supersc r ip t denotes the expected value condi t iona l on information ava i l ab le at time t . Thus K*g i s spec i f i ed as a funct ion of the expected paths of a l l exogenous va r i ab les cond i t iona l on information at time t : K * t = K * * ( P £ . , Y * , U * s ) = K**(x*), where x s i s a vector of a l l exogenous va r i ab les at time s. Note that the cha rac te r i za t ion o f t h i s expression as a d i s t r i bu t ed lead equation with geometr ica l ly dec l i n i ng weights fol lows from the assumption at each time per iod of a constant rate of discount r and an adjustment parameter x which make up the 'augmented' d iscount rate (1-x+r). Thus the d i s t r i bu t ed lead form resu l t s d i r e c t l y from the s p e c i f i c a t i o n of the model rather than j u s t a " f e e l i n g " that the weights dec l ine smoothly. As in the standard geometr ica l ly dec l in ing lag formulat ion, i f one attempted d i r e c t l y to estimate (2 .4 .10) , problems would r e s u l t from the lack of degrees of freedom. -131-The expectat ions process i s assumed to be charac ter ized by ce r ta in ty or ce r t a in t y equivalence and therefore " r a t i o n a l i t y " , so that at time t , f o r any exogenous var iab le ir and fo r any time s, ir g = ir^ + e^ with E . e^=0 . Thus e* i s the e r ror in expectat ion of ir, formed at time t . t s s r s The subscr ip t t on the s t a t i s t i c a l expectat ion term ind ica tes that th i s expectat ion i s cond i t iona l on information ava i l ab le at time t . This s p e c i f i c a t i o n i s c r u c i a l fo r the Koyck adjustment formulat ion in terms of l eads , f o r with unobservable future var iab les the actual i ncor rec t p red i c t ion of the exogenous var iab les by the f i rm must be taken into account ex post . In other words, expectat ions at time t and time t+1 instead of j u s t at time t are re levant in the lead formulat ion and the extent to which they d i f f e r depends on the magnitude of the e r ro r in expectat ions in time t fo r per iod t+1. This i s accommodated in time t+1 by adjust ing the path of investment from that which was determined by the values of exogenous var iab les experienced in the previous time pe r iod . Such a representat ion incorporates the "genera l " expectat ions form d i r e c t l y in to the model by al lowing observed values of the exogenous va r i ab les plus some form of e r rors in expectat ions in that per iod to r e f l e c t the expectat ions process. The Koyck adjustment process for d i s t r i bu t ed leads i s thus s l i g h t l y d i f f e r en t than that for lags because expected future var iab les are unobservable even ex post , which impl ies a problem of adjustment of the independent var iab les with the t ransformat ion. To respec i fy the equation in terms of observable I T ' S , subst i tu te ifg-e^ fo r TT^ d i r e c t l y and subs t i tu te K* -e j * fo r K** f in (2.4.10) to ob ta in : 2.4.11) AK t = x K t _ x - xK* t + (1/Y^) S s * t ( l / ( l - x + r ) s " t + 1 ) ( ( Y K K + r Y j ^ K ) ( K * s - K * t ) + Y E K A P E S + Y M K A P M S + Y Y K A Y S -132-- « l ^ k k > ' 5 3 ^ 1 : ( X / ( } S " t + 1 } ( ( ^ K K + r ^ K ) e K * s + Y EK e Es + YMK eMs + Y Y K e Y s ) ' To reduce the number of observables (the exogenous var iab les on the r i gh t hand s ide of (2 .4 .11) ) , the equation i s transformed by leading once, and mu l t ip l y ing by ( l - x + r ) - 1 , y i e l d i n g : 2.4.12) AK t + 1 / ( l -x+r ) = (x/( l-x+r) )K t - ( x/ ( l -x+r ) )K* t + 1 + (l/( l-x+r) Y |^) Esn+1 ( 1 / ( 1 - ^ ) S ' t , ( ( ^ % ) ( K V K V l » + Y E K A P E S + YMK* PMs + M A Y s ) - ( l / ( l - x + r ) Y ^ ) £ - t + 1 ( l / d - x + D ^ J t t y ^ r r ^ l e ^ . . t + 1 . , t+1 . .t+1, + YEK Es + YMK Ms + YYK Ys h Subtract t h i s from equation (2.4.11) (note that e£*j=0 (Abel)) to de r i ve : 2.4.13) AK t = ( l / ( l -x+r ) )AK t + 1 + x K ^ - (x/( l-x+r) )K t - xK* t + ( x/ ( l -x+r ) )K* t + 1 + (l/y^)H"=t ( 1 / ( 1 ^ + r ) S " t + 1 ) ( ^ K K + r Y k K ) { K * t + r K * t ) + U / d - x + r h j ^ ) ( Y E k A P E t + Y Mk A P Mt + Y Y K A V " B r = t + 1 d / ( l - ^ ) S - t + 1 ) ( ( Y K K + r Y k K ) (pt - e t + 1 ) + Y - (e^ -e* + 1 ) + Y ' ( e £ -e^ + 1 ) + l e K * s eK*s' YMK Ms eMs ' YEK Es e E s ' YYK Ys e Ys Since K t= A K t + K t _ ^ , t h i s becomes: 2.4.14) AK t ( ( l+r )/ ( l-x+r) ) = ( l / ( l -x+r ) )AK t + 1 - ( ( x 2 - rx ) / ( l -x+r ) J K^ + ( ( x 2 - r x ) / ( l - x + r))K*. + (l/( l-x+r) ) (Y^^AP^ .^ + Y ^ A P ^ + Yy^AY^) -133-, t t+1, . , t t+1 \ , , t t + l v * + YMK Ms" Ms 1 + YEK Es" Es } + YYK Ys" Ys ] ) > so, subs t i tu t ing fo r K* t +^ and K* t : 2.4.15) AK t = A K t + 1 / ( l + r ) - ( ( x 2 - r x ) / ( l+ r ) ) K t _ 1 - ( ( x 2 - r x ) / ( l+ r ) Y | < R ) { a K + Y E K P E t + Y M K P M t + a K t t + Y Y K Y t + U K t ) + ( l / { 1 + rHk ] ^ ~ t + 1 ( l / ( 1 - X + r ) 5 " * ) ^ ^ Y K K + r Y K K ^ Y K 0 ^ E K ^ E s ' ^ s ^ + W e M s " e M s 1 ) , ,t t + l v , / t t + l v x , t t+lx + YYK Ys Ys ' + ( e K s " e K s } ) " YMK Ms" Ms } . y > ( e t _ e t + 1 ) _ Y -(e^ -e* + 1 ) ) EK Es e E s ' YK Ys e Ys where A P . t = ( P u - P . ^ , A Y ^ Y ^ Y ^ ) , ^ K V ^ K r t . t - l * ' a n d e E s = P E s " P E s ' 4 = P M s " P M s ' 4s= Ys" Ys' a n d e K s = U K s - u K s ' impl ied from e** s =K* s -K*^ as well as e ^ A P ^ - A P ^ , eJ^AP^-AP^ , e£ =AY =AY t from above. Ys s s Note that s ince there i s no s tochast i c s p e c i f i c a t i o n on the o r i g i na l equat ion, only problems r esu l t i ng from the determin is t i c s p e c i f i c a t i o n are r e f l e c t ed in th i s equat ion. More s p e c i f i c a l l y , i f a c lean disturbance term were appended to the o r ig ina l equation before the transform, the transform would induce se r i a l co r r e l a t i on (moving average dependence) on the disturbances in the f i n a l equat ion, s im i l a r to the t r ad i t i ona l r e su l t on a simple adaptive expectat ions model transformed by a Koyck adjustment procedure. Note, however, that t h i s s t ructure does not depend on the form of the expectat ions formation assumption but rather depends on the determination of future revenue by an i n f i n i t e path of expected va lues. This s tochast ic -134-cha rac t e r i za t i on i s not taken in to account here; inc lud ing i t causes fur ther s p e c i f i c a t i o n and est imat ion d i f f i c u l t i e s , fo r se r i a l c o r r e l a t i o n plus the lead dependent var iab le and ' innovat ion ' terms e ^ could r e su l t in i n e f f i c i e n t or incons i s ten t estimates with t r ad i t i ona l est imat ion methods. Another d i f f i c u l t y a l luded to above i s the i n te rp re ta t i on of the e ^ terms. Unl ike the Koyck lag case where a l l past values are g iven, observable and therefore cance l , in th i s case they cancel only i f the expected values of future exogenous var iab les do not change with new in format ion, i . e . , i f expectat ions at time t of va r iab les at time t+1 were per fec t l y determined ex post . The economic agent i s expected to adapt his expectat ions in per iod t+1 given mistakes from time t that become evident in time period t+1. To i l l u s t r a t e t h i s i t i s useful to attempt to i n t e rp re t the i n f i n i t e sum depending on the e r ror terms, e ^ . By const ruct ion of equation (2 .4 .15 ) , i t i s evident that e ^ - e ^ 1 = T T ^ + 1 - ir*. Putt ing the exogenous var iab les e x p l i c i t l y in t h i s form, one obta ins : a) t e K s " e Ks = t+1 U Ks " t U Ks« b) e Ms" e t + 1 eMs = pt+1 Ms r Ms» c) e t + 1 e E s = p t + l K Es " 4 s -d) • r V e t + l e Y s = y t + l _ s e) Ms e t + l eMs = < - < • f ) *EV e E s = - 4 P E s -g) e Y s = In other words, "the change in the expectat ional e r ro r from per iod t to per iod t+1 i s equal to the change in the expectat ions from per iod t to per iod t+1. -135-This fo l lows from the fac t that in every pe r i od , the expectat ion of some future var iab le plus the expectat ional e r ro r i s equal to i t s subsequent 37 r e a l i z a t i o n " . If the economic agent 's expectat ions were ce r t a in and c o r r e c t , then t h i s term would cance l . However, in t h i s model the agent i s e i the r c e r t a i n about h is expected path and never learns although he i s wrong every yea r , o r , there i s an e r ro r but i t i s 'white n o i s e ' . The l a t t e r assumption i s cons i s ten t with the model. Combining (2.4.15) and (2.4.16) and s imp l i f y ing one can de r i ve : 2.4.17) ( l + r ) A K T - A K T + 1 = - x ( x - r ) ( K T _ 1 + ( l / Y K K ^ a K + Y E K P E t + Y M K P M t + a K t t + Y Y K Y t + U K t ) } + ( 1 / ^ ) ( Y E f c A P E t + Y M K A P M t + T Y K A Y t ) + R t + 1 ' where, R t + 1 - d / ( l ^ ) ^ ) £ s ; + 1 ( l / d - x + r j ^ ^ f t r ^ r r ^ ) / ^ ) + ( U ^ - l i J s ) ) - ( A P j ^ - A P J s ) + T £ | e ( A P ^ - A P ^ ) + Y Y K ( A Y s t + 1 - A Y s ) } -Thus, R t + 1 i s equal to the discounted value at per iod t of the sum of rev i s ions in expectat ions for a l l exogenous va r i ab l e s , weighted by the respect ive c o e f f i c i e n t s , based on information which becomes ava i l ab le in per iod t+1 but was not known in per iod t . Note, however, that R t + 1 i s based on innovations in per iod t+1, and thus can be in te rpre ted as a composite disturbance term which contains an unobservable omitted var iab le co r re l a ted with A K ^ + 1 » Therefore , i f there were a parameter appended to A K t + 1 , est imat ion by ordinary l eas t squares or even conventional maximum l i k e l i h o o d would not r e su l t in cons is tent -136-est imates . However, in t h i s case , in cont ras t to A b e l ' s model, the A K ^ + 1 term i s simply m u l t i p l i e d by data so that i t can be added on to the l e f t hand side of the est imat ing equat ion, rather than on the r i gh t as in (2 .4 .17 ) , r e su l t i ng in c o r r e l a t i o n between the dependent var iab le and the composite disturbance term. Th is avoids the potent ia l problem of incons i s ten t est imates. In terpreta t ion of the A K ^ term i s informative and f a c i l i t a t e s understanding of the f i na l model. As Abel (1978) points out, the lead dependent va r i ab le before the transform, in th i s case A K T + 1 , may be large for two d i f f e r e n t reasons: i ) op t im i s t i c expectat ions of future exogenous var iab les as of time t as well as at time t+1, which would r e su l t in a high value o f A K T + 1 , as well as a high value of the current va r iab le A K T , and i i ) upward rev i s ions in expectat ions in per iod t+1, which would cause A K T + 1 to be large but should not be r e f l e c t ed in A K T . This i s the way in which the 'supplementary' disturbance term R t +^ (the discounted sum of rev i s ions ) enters in to the a n a l y s i s . I t causes A K T to be lower for a given A K T + 1 i f the high value of A K T + 1 i s caused by information which was ava i l ab l e at time t+1 but not at time t . This de r i va t ion has impl i ca t ions for the in te rp re ta t ion of the transformed dependent va r i ab le in the model, A K ^ - ( l / ( l + r ) ) A K T + 1 . Note that l / ( l + r ) A K T + 1 i s equiva lent to de f l a t i ng A K T + 1 to be comparable to A K T in cur rent va lue , so that i t becomes the cur rent valued f i r s t d i f f e rence in investment. Th is impl ies that the dependent var iab le w i l l be large and pos i t i v e (negative) i f expectat ions are rev ised downward (upward) in per iod t+1 so that A K T + 1 i s small ( large) r e l a t i v e to A K ^ . . It remains to spec i fy R t+^ in terms of observable va r i ab l e s . R t +^ depends on the change in the information set when i s revea led. However, s ince a l l i r £ ^ are based on the information inc luded in ^t+l w n i c h 1 S observable as i r t + 1 , i T t + 1 incorporates the new -137-information on which R ^ i s based. Thus can be used as a proxy f o r Rt+1 l f T t 1 S 9 e n e r a t e d b y a n ARMA(p.q) process. Abel (1978) showed t h i s fo r an AR(p) process. His ana lys is can be genera l i zed , however, to any covariance s tat ionary ARMA(p,q) process since such models y i e l d an analogous formulat ion with a ' r a t i o n a l ' lag s t ruc tu re . Abel shows that , i f i r t =#(L ) i r t +n t , where d(L)=X)i=i tf-jL1 i s a lag polynomial of order p, n^ . i s white no ise , and expectat ions are formed r a t i o n a l l y , one can de r i ve : 2.4.18) R t + 1 = ( Y / ( l - « J ( Y ) ) ) ( l - r f ( U K t + 1 , where y i s equiva lent to the discount rate l/(l-x+r) and in th i s case i s a vector of exogenous var iab les observed at time t+1, weighted by t he i r respect ive c o e f f i c i e n t s . Thus d R ^ / d i r ^ X ) , implying that a high value o f i r t + ^ i s assoc iated with a high value of R t + ^« S i m i l a r l y , in the more general ARMA (p,q) case : 2.4.19) i r t = (d(L)/e(L ) ) i r t + n t , where e(L) i s equal to E j ^ e - j L . 1 . This polynomial i s therefore ' r a t i o n a l ' and tf(L)/e(L) can be rewr i t ten as tf'(L), which replaces #(L) in the c a l c u l a t i o n s . Such a procedure l i n k i n g the spec i f i c a t i on of the determin is t i c model with the " r a t i o n a l " expectat ions assumption discussed above resu l t s in an equation with the c o e f f i c i e n t on the proxy T r t + 1 equal to a funct ion of y, ${\-)/i{y) and the c o e f f i c i e n t s corresponding to each exogenous var iab le i n d i v i d u a l l y . In th i s 'genera l ' case , components of the f i n a l parameters cannot therefore be separately i d e n t i f i e d except fo r those estimated independently i n the equat ion, but the composite c o e f f i c i e n t estimate can be determined. -138-Thus the determin is t i c est imat ing form fo r the investment equation i s : 2.4.20) ( l + r ) A K t - A K t + 1 = - x U - r M K ^ + (1/ T | ( K ) ^ o t K + Y EK' 5 Et + Y MK' > Mt + a Kt^ + T Y K Y t + u K t ) ) + ( 1 / Y k k ) ( Y E k A P E t + Y M k A P M t + Y Y K A Y t ) + u y Y t + l + u E P E , t + l + "M P M,t+l + u K u K , t + l + u E A P E , t + l + "M A P M,t+ l + u Y A Y t + l ) * In th i s formulat ion there are no expected va r i ab l e s ; the actual values at ta ined in per iod t+1 are j u s t subs t i tu ted . As discussed above, t h i s takes into account the expectat ions process, but the parameters of the expectat ions process cannot be ca l cu la ted within th i s formulat ion. Even though the ind iv idua l parameters cannot be i d e n t i f i e d , some of the composite c o e f f i c i e n t s can be shown to have i n t u i t i v e l y determin is t i c s igns . It i s not en t i r e l y c l e a r what sign the c o e f f i c i e n t s on v t + ^ , P £ t + ^ , P M t+1' a n d UK t+1 m u s t b e ' D e c a u s e t he i r e f f e c t on the change in investment from time per iod t to t+1 depends on the i r r e l a t i onsh ip to the analogous var iab les in the previous pe r iod . Thus the remaining va r i ab l e s , A Y t + l ' a P E t+1' A N D A P M t+1 ' i a v e c l e a re r imp l i ca t ions . fo r example, should have a negative c o e f f i c i e n t u y s ince i f Y i s inc reas ing , ^t+1 w o u ^ b e e x P e c t e d to be la rger than K^, r e su l t i ng in a small ( l+ r )K t -K t + ^ . The c o e f f i c i e n t s on A P ^ and A P ^ depend on the i r s u b s t i t u t a b i l i t y or complementarity with the cap i t a l input ; i f they are complements, increas ing the pr i ce impl ies a lower desired cap i t a l stock and thus a lower AK t + ^ than A K ^ , r e su l t i ng in a pos i t i v e c o e f f i c i e n t . For subs t i tu tes , the reverse holds. F i n a l l y , note that the adjustment parameter x can be ca l cu la ted e x p l i c i t l y as the root of the homogeneous part of (2 .3 .11) , r e su l t i ng i n : 2.4.21) | x | = - ( l /2 ) ( r- ( r 2 + 4 ( T K K + r Y ^ K ) / T | ^ ) - ( 1 / 2 ) ) . -139-This expression of the adjustment c o e f f i c i e n t as a funct ion of r and the technologica l c o e f f i c i e n t s in G i s subst i tu ted in to (2.4.20) to der ive the actual f i n a l est imat ing equat ion. With the Koyck transformation u t i l i z e d to const ruct t h i s equat ion, however, one add i t iona l d i f f i c u l t y a r i s e s . A supplementary disturbance term ex i s t s conta in ing terms which are co r re l a ted with i r t and i r t + j unless TT^ i s generated by an ARMA process that has s e r i a l co r r e l a t i on going back only one pe r i od , i . e . , A R ( 1 ) , MA(1 ) , or A R M A ( 1 , 1 ) . 3 8 This i s the case because the c o e f f i c i e n t on i r t + p Y (1-^(D )/(1-«HY) ), can be broken down in to two components, Y/(1-«KY)) and Y«J*(L)/(1-0(Y) ), where tf*(L) i s def ined as of the disturbance term. In general t h i s w i l l be co r re l a ted with ir^ and i r t + j . However, i f the se r i a l c o r r e l a t i o n process only inc ludes consecut ive per iods , then / S * ( L ) i r t + ^=0. In the former case one must carry out est imat ion using instruments which are co r re l a ted with ir .^ and T r t + ^» but not with 39 ^ t - l ' ^ t-2 ' ^ o r t ' i e ' ' a t t e r a r e components of the e r ro r term. This impl ies estimates are cons i s ten t i f expectat ions are formed by an adaptive expectat ions process character ized by an IMA( l , l ) model; i . e . , i f i s charac ter ized by an IMA(1,1) process, AP t can be spec i f i ed by a MA(1) process, cons is tent with the above argument. I f , however, evidence of s e r i a l c o r r e l a t i o n ex i s t s going back two periods or more, t h i s equa l i t y w i l l not ho ld . The above adaptive expectat ions s p e c i f i c a t i o n therefore ensures cons i s ten t est imates , and allows i d e n t i f i c a t i o n of s t ruc tura l expectat ions c o e f f i c i e n t s . Unfortunate ly , i t i s a lso more r e s t r i c t i v e in terms of assumptions on the expectat ions formation process and thus i s l i k e l y to conta in some m i s s p e c i f i c a t i o n . Given an adaptive expectations s t ruc tu re , the e x p l i c i t form of the change in the path of future expectat ions can be ca l cu la ted using knowledge about t+1 d.L . This second component, mu l t i p l i ed by ir t+1' becomes part -140-revealed in the r e a l i z a t i o n of t+1 va lues . The concept underly ing t h i s der i va t ion i s that any future expected value at time t , i r * + s (s>0) depends on r ea l i z ed values of (k=0,...«>) and expected values ""t+i ( i ^ * * * * s-1). Thus the expected value of ir** 1 w i l l d i f f e r from i r * + s by the impact of the information gained in per iod t + i , charac ter ized by t+1 t t t , . . ^t+l'^t+l = ^t+l'^t+l = e t + l ' r ea l i zed e r ro r in the t per iod forecast of the (t+1) expected value. It i s not c l e a r that t h i s captures the ent i re impact, however, s ince the expectat ions process i s adaptive so that t h i s e r ro r a lso a f f e c t s subsequent expected va lues . To der ive how the impact would be d i s t r i bu t ed in th i s s p e c i f i c case , ir* and irpT1 can be wr i t ten out to determine the d i f fe rence t+s t+s a s s _ 2 , 3 , . . . : 2.4.22a) T T £ + 2 = ( 1 - B ) ( i r * + 1 + Birt + B 2 * ^ + ...) t+1 2 b) i T £ + 2 = (l-B)(ir^+2 + BTTJ. + B ir^._^ + . . . ) , SO, ^ t + 2 " ¥ t + 2 = ( 1 _ B ) ( i r t + r i r t + i ) = ( 1 _ B ) e t + l ' where B i s the adaptive expectat ions c o e f f i c i e n t . S i m i l a r l y : 2.4.23a) TT£ + 3 = U-B )U* + 2 + Bir£+1 + B 2ir t + ...) b) TT^2 = ^ " B ^ i r t + 2 + e i r t+l + ^ t + SO, ir**3 - i r * + 3 = ( l - B ) ( i r ^ 2 _ i r t + 2 ) + 6 ( 1 _ B ) { ^t+ l^t+ l ) = (1-B)e* + 1 + B(l-B)e*+ 1 = ( l-B ) e * + r If one extends th i s analogously to a l l irj;*1 - u* + one f inds that the r ea l i zed e r ro r in per iod t+1 s h i f t s upward the level of th i s en t i r e path of expectat ions by (1-B ) e£ + 1 . Thus, a f t e r the f i r s t shock the -141-changes in expected p r i ces are expected to be inva r i an t . The en t i r e impact of the r e a l i z a t i o n of t+1 values can be charac ter ized as ( 1 - B ) e t + 1 and subst i tu ted in to the est imat ion process as the actual i r t + 1 value minus the Tt+1 v a ^ u e ca l cu la ted by an adaptive expectat ions or IMA(1,1) process. The above r e su l t i s cons i s ten t with the d iscuss ion in the previous sect ion on the i n t e rp re ta t i on of the IMA(1,1) as an adaptive expectat ions process. Equation A10 in Appendix A charac te r izes the IMA(1,1) process, reproduced here in the current no ta t ion . From th i s one can see more c l e a r l y in what sense the impact ( 1 - B ) e t + ^ can be in te rp re ted : 2.4.24) i r t + 1 = w t + (1-B)e t . It i s t r ad i t i ona l in the l i t e r a t u r e to use th i s equation to i n t e rp re t ( l-B)e as the 'permanent' con t r ibu t ion of e t and Be t as the ' t r a n s i t o r y ' c o n t r i b u t i o n . 4 0 Such an i n t e rp re ta t i on i s cons is tent with the idea that the permanent change in expectat ions r e su l t i ng from one pe r i od ' s more information ( e t here) on a future expected value i s r e f l e c t ed by ( l - e ) e t . Thus, assuming that the expectat ions process can be modeled by an adaptive expectat ions model or an IMA{1,1) process on the l eve l s of expected var iab les in the e q u a t i o n , 4 1 I subst i tu te in to equation (2.4.17) to der ive the est imat ing equation with adaptive expectat ions : 2.4.25) ( l + r ) A K t - A K t + 1 = - x U - r M K ^ + U / Y K K M V W ^ t + ^ E ^ E . t + l ) + W P M t + ( 1 - B M ) e M , t + l ) + ^ + ( I / Y K K ) ^ E K ( A P E ^ + Y Y K ( A Y t - ( l - B Y ) e Y } t + 1 ) ) . -142-It i s c l e a r that in t h i s formulat ion the expectat ions process must be ca l cu l a t ed in order to der ive both the B ' S and the e t + 1 ' s . Thus the s i m p l i f i c a t i o n r e su l t i ng from Abe l ' s procedure, which bypasses th i s s tep , i s not app l i c ab l e . However, the expectat ions parameters in t h i s case can be e x p l i c i t l y i d e n t i f i e d and eas i l y i n te rp re ted . S p e c i f i c a l l y , the parameters are estimated separately from the IMA(1,1) process and once the r esu l t i ng numbers are subst i tuted i n , the current p r i ce i s in e f f e c t " de f l a t ed " by these values to take into account the discounted future expectat ions of exogenous v a r i ab l e s . In add i t i on , i t i s important to note that although t h i s expression appears complex, the f a c t that est imation i s c a r r i ed out in two steps subs tan t i a l l y s i m p l i f i e s matters. Once the parameters of the expectat ions process and r e a l i z ed er rors are der ived and subs t i tu ted , the f i n a l est imat ing 42 equation reduces to a t rac tab le formula t ion . Recal l that inconsistency of estimates may be induced by subs t i tu t ion of a complex lag s t ructure r e su l t i ng in e r rors in the disturbance term co r re l a t ed with the regressor subst i tu ted fo r R t +^' A complex se r i a l co r r e l a t i on s p e c i f i c a t i o n would a lso cause e t + ^ to be co r re l a ted with past disturbances which would become components of the e r ro r term. Then the co r r e l a t i on between the regressor (T^) and th i s supplementary component of the disturbance term w i l l cause b i a s , even asymptot i ca l l y , and should be taken in to account unless the bias can be shown to be sma l l . However, with the IMA(1,1) process assumed, that problem i s avoided, f o r s e r i a l co r r e l a t i on i s important only fo r consecutive per iods . Otherwise, to obtain cons i s ten t estimates with these models with no se r i a l c o r r e l a t i o n term from the o r i g i na l d i s t u r b a n c e s , ^ se r i a l c o r r e l a t i o n should be purged from the omitted part of the R t + 1 term which w i l l be in the disturbance term and be cor re l a ted with the proxy i r t + 1 or with e t + ^ . If t h i s i s not the case , however, s t r i c t l y speaking se r i a l c o r r e l a t i o n w i l l -143-l i k e l y e x i s t , and should be dea l t with by instrumental va r iab les which should be co r re l a t ed with ir t and i r t + ^ but not with lagged values of ir. I f future va r i ab les such as are used fo r instruments, however, degrees of freedom are l o s t . In add i t i on , problems r e su l t i ng from se r i a l co re l a t i on may be neg l i g i b l e i f s e r i a l co re l a t i on dies out f a i r l y qu i ck l y . I.e., "In p r a c t i c e , the c o e f f i c i e n t s of the lag polynomial of order two or higher may be small enough so that incons is tency due to the co r r e l a t i on of ir t and i r t + 1 with 44 0 * ( L ) T r t + i may be small . F i n a l l y , although instrumental var iab les methods are cons i s t en t , in t h i s case they are s t i l l biased so that in small samples i t i s not c l e a r whether t h i s method i s more des i r ab l e . Replacment of the unobserved R t +^ by i r t + ^ (or e t + ^ in the adaptive expectat ions case) may be adequate, at l e as t as an approximation, to mit igate problems of incons is tency . Using these r a t i o n a l i z a t i o n s I estimate the equation under the assumption of a white noise e r ro r s t ructure on the f i n a l investment equat ion, assuming that any ' r e s i d u a l ' c o r r e l a t i on from subs t i tu t ing i r t + ^ for the unobserved R t + 1 term i s neg l i g i b l e in the general form and that no incons is tency e x i s t s in the adaptive expectat ions form. Th is i s tes tab le in a "crude" way by attempting an a l t e rna t i ve instrumental va r iab les s p e c i f i c a t i o n and comparing r e s u l t s . It might be noted here that empir ica l r e su l t s of these types of tes ts ind ica ted that even with the loss of information from one more important recent data year , l i t t l e change occurred with the instrumental va r i ab les procedure. To implement th i s s p e c i f i c a t i o n , parameters of the system of va r iab le input demand equations and the quas i- f ixed input accumulation equat ion, (2 .4 .5 ) , (2 .4 .6 ) , ( 2 .4 .7 ) , and (2.4.20) or (2.4.25) for the adaptive expectat ions case , are estimated by maximum l i k e l i h o o d under the assumption that a disturbance term can be added to each equat ion, and that the r e su l t i ng -144-disturbance vector i s independently and i d e n t i c a l l y mu l t i va r i a te normally d i s t r i bu t ed with mean vector zero and constant nonsingular covariance matrix W. To incorporate the general expectat ions formations model in to equation 2 .4 .21, one does not need to subs t i tu te r ea l i z ed values of the expectat ions formations process . However, to impose the adaptive expectat ions mechanism on (2.4.25) the expectat ions formation process must be ca l cu la ted to obtain estimates of the current per iod e r ro r or ' i nnova t ion ' and of the expectat ions adjustment c o e f f i c i e n t to subst i tu te in the f i na l investment equat ion. Th is i s accomplished using maximum l i k e l i h o o d ARIMA algorithms by spec i f y ing an IMA(1,1) process on the va r i ab l es . More complex ARIMA(p,d,q) models are then estimated to determine i f an adaptive expectat ions procedure or other method with one-period se r i a l co re l a t i on i s an adequate representat ion of the expectat ions process. This f a c i l i t a t e s determining consistency and a lso i n t e rp re t i ng c o e f f i c i e n t s . Once t h i s model i s est imated, the c h a r a c t e r i s t i c s of the production process can be summarized by short run, myopic long run and long run pr i ce and output e l a s t i c i t i e s . Short and myopic long run e l a s t i c i t i e s are def ined 45 s i m i l a r l y to short and long run e l a s t i c i t i e s in previous s tud ies . However, in t h i s case the " t rue" long run i s not as e a s i l y cha rac te r i zed . Short run e l a s t i c i t i e s are def ined as va r i a t ions in "ad jus tab le " f ac tors given exogenous changes with cap i t a l f i x e d . The long run i s def ined as that occurr ing when a l l adjustment toward the "des i r ed " cap i t a l stock has been completed. Th is can now be def ined in two a l t e rna t i ve ways corresponding to two d i s t i n c t i n te rp re ta t ions of the " long run" des i red cap i t a l s tock: 1) as K*(t ) , the "des i red " cap i t a l stock given current exogenous va r i ab l e s , or 2) as a composite K** inc lud ing K*(t) and the changes in the des i red cap i t a l stock corresponding to expected changes in exogenous va r i ab les . I c a l l the former the myopic long run and the l a t t e r I charac te r ize as the " t rue " long run. -145-Confusion can e a s i l y a r i s e , however, because with a varying path of future exogenous va r i ab l es , the f irm never reaches a " long run" K*, for the target depends on t . Thus the long run e l a s t i c i t y i s an ambiguous concept to i n t e rp r e t . For a representat ive case of the own pr i ce e l a s t i c i t i e s of demand for energy the myopic long run and long run e l a s t i c i t i e s are represented respec t i ve l y as : 2.4.26) e M ^ e = (P E t/E)(3E/3P E t) /K=K t + 3Et/3K*t {aK*./3PEt), or 2.4.27) e ^ p e = ( P E t / E ) ( 3 E / 3 P E t ) | K=K t + 3E t / a K * t (3K** t/3P E t)* The f i r s t cha rac te r i za t i on of the long run e l a s t i c i t y corresponds to previous s p e c i f i c a t i o n s and r e l i e s on the der i va t i ves of K* t . These are f a i r l y simple given the form of K* t , reproduced here: 2.4.28) K* t = (-1/Y k k ) (- k + Y E K P E t + YMK PMt + "iCt* + Y Y K Y t + \t ]' So that 9 K * t / 3 P k t ' -\K/\K> *nd 4"pe = P E t / E t ( Y E E " Y E K / Y K K K The de r i va t i ves of K** t are not nearly as s t ra ight forward , fo r they r e l y on the en t i r e opt imizat ion process over t ime. To obtain a grasp on how the d i f f e rence can be in terpre ted and a n a l y t i c a l l y der i ved , r e ca l l from the in t roduct ion that the form of the investment equation fo r the s t a t i c and non-stat ic expectat ions cases can be expressed as: 2.4.29) (a) AK t = x (K* - K ^ ) , and (b) AK^. = | x | ( K * * t - K t _ x ) = x K t _ 1 - x K * * t , r e spec t i v e l y . The i n t e rp re t i on of the l a t t e r case i s that the f i rm i s aiming at a " des i r ed " cap i t a l stock K * * in the intertemporal sense by taking the present value of a l l future costs and benef i ts into account and balancing them -146-in i t s current adjustment dec i s i ons . From (2.3.21) and i t s d i sc re te approximation (2 .3 .32) , K**(t) can be restated as : 2.4.30) K*(t) + J = K*(t) + U / x E ~ t ( l / ( l - x + r ) s _ t + 1 ) ( Y ^ Y ^ / Y ^ (K*(s)-K*(t)) + G R p P (s ) + G^ Y Y(s) ) . o From 2.3.19, ( Y R R + I " Y R R ) / Y r r = _ b = (x -rx). Thus (2.4.29b) becomes, 2.4.31) K**(t) = K*(t) + ( x - r E ^ t l/(l- x + r ) S _ t ( K * ( s ) - K * ( t ) + G^P(s ) + G^yYfs). Th is equation i s useful f o r obta in ing an i n t u i t i v e idea of what the long run e l a s t i c i t i e s mean. The der i va t i ve of K**(t) with respect to an exogenous var iab le i s the change in the current des i red cap i t a l stock inc lud ing the shock, plus a weighted sum of i t s e f f e c t on the future des i red cap i t a l stocks in terms of the path of future expected values of the exogenous v a r i ab l e s . Unfortunate ly , i t i s d i f f i c u l t to determine the l a t t e r e f f e c t a n a l y t i c a l l y from th i s express ion, s ince the form of the expected changes over time i s not evident e x p l i c i t l y from the equat ion. It i s more u s e f u l , although less i n t u i t i v e , to determine the e x p l i c i t expression fo r the der i va t i ve from the transformed investment demand equation (2 .4 .20) . This can be accomplished by wr i t ing out the f i n a l investment equation (2.4.20) in terms of (2 .4 .29b) : 2.4.32) (l+ r ) A K t - A K t + 1 = -x(x-r) (K t _ r (K* t +J) ) - - x f x - r H K ^ - K* t + ( l / ( A 2 - r x ) ) ( l / Y K K ( Y E K A P E T + Y M R A P M T + Y Y ^ Y t ) + u y Y t + 1 + ( J E P E , t + l + u m P M , t + l + u K P K , t + l + " E A P E , t + u M A P M , t * l + u Y A W > where (K*.+J) i s K**, and J i s : -147-2.4.35) J - - ( l /U 2 -rx ) ) ( l / r ,< K (^^ u E P E J t n + V M , t n + u K P K , t + l + u E A P E , t + " M A P M , t + This s p e c i f i c a t i o n provides a useful way to expand the formula fo r the der i va t i ves to incorporate the extra adjustment corresponding to non-stat ic expectat ions. In t h i s way der i va t i ves of K** can be e x p l i c i t l y spec i f i ed as a funct ion of de r i va t i ves of K* and J , and the impact of the non-stat ic expectat ions process can be i s o l a t e d . Th is i s very use fu l , for i t al lows one to explore the re l a t i onsh ip between the adjustment corresponding to current va r i ab l e s , represented by K*(t ) , and the fur ther adjustment made from expectat ions. C lea r l y the e l a s t i c i t i e s r e su l t i ng from th i s s p e c i f i c a t i o n are more complex than the "convent iona l " long run e l a s t i c i t i e s . It i s useful therefore to present the energy e l a s t i c i t y from (2.4.27) as a representat ive e l a s t i c i t y formula. From (2 .4 .27) , the required expression to ca l cu l a te i s 3 K * * t / a P E t . Using (2.4.32) and (2.4.33) t h i s can be ca l cu la ted as : 2.4.36) aK* t/3P E t + 3 J / 3 P E t = 3K* t /3P E t - ( l/(x 2-rx)) W^A,^-^), = ( Y E K / Y K K + l / ( x 2 - rx ) (Y E K/YKK- "E ) K Since 3 E t / 3 P E t and 3E t / sK* t can be derived d i r e c t l y from (2 .4 .5 ) , the f u l l long run own e l a s t i c i t y f o r E becomes: L' 2 2.4.37) e E p e = P E t / E t ( Y E E " Y E K ( W Y K K + 1 / ( x " r x ) { Y E k / T k k _ 0 E ) ) * Other e l a s t i c i t i e s depending on more complex equations a re , of course, more compl icated, but the procedures used to ca l cu l a t e them are analogous to those above. Note that the st ructure of these models impl ies a s l i g h t l y d i f f e r en t i n t e rp re ta t i on of the adjustment c o e f f i c i e n t | x | than in previous work, in -148-which the parameter ind ica ted the proport ion of f i n a l adjustment toward the long run state that the f i rm would pursue. In the current context the f i rm w i l l be adjust ing at a d i f f e r e n t rate than i s j u s t i f i e d given the cur rent economic c l imate i t perce ives ; s p e c i f i c a l l y , i t w i l l exh ib i t extra adjustment in response to future costs and benef i ts from future changes in exogenous va r i ab l e s . In t h i s sense the c o e f f i c i e n t s on the changing exogenous var iab les can be in terpre ted in terms of in terna l costs of adjustment; the f i rm w i l l incur extra current costs up to the po int where future returns w i l l compensate. In other words, i f the pr i ce of a va r iab le input , say l abor , i s increas ing ( r e l a t i v e to the discount r a t e ) , then the costs of using th i s labor next per iod to f inance cap i t a l adjustment w i l l be even higher than today. Thus more adjustment w i l l be c a r r i ed out t h i s pe r i od , even i f the current s i t ua t i on does not warrant t h i s extra co s t . S i m i l a r l y , i f the f i rm expects future output to be i n c reas ing , then the potent ia l present value p r o f i t from a un i t of cap i t a l now increases , o r , converse ly , the marginal cos t of the cap i t a l cons t r a in t at a given cap i t a l leve l increases . This impl ies that the f i rm w i l l want to increase current cap i t a l stock to correspond more c l o se l y to expected output l e v e l s ; capac i ty output w i l l not lag as fa r behind as with 45 naive expectat ions . Thus, given expectat ions of an expanding economy ( implying increas ing output and pr ices ) the f i rm w i l l have a l a rger perceived gap between Kt_^ and i t s "des i r ed " cap i t a l stock K**, causing the absolute amount o f adjustment to be higher, even though the corresponding proport ion may be smaller s ince the proport ion i s in terms of a l a rger "gap" . Now that the non-stat ic expectat ions model and the r e su l t i ng charac te r i za t ions of the technology represented by the e l a s t i c i t y formulas have been der i ved , in the next sect ion I turn to a d iscuss ion of the r esu l t s of empir ica l es t imat ion . V. Empir ical Results -149-In t h i s sec t ion I report empir ica l r e su l t s from maximum l i k e l i h o o d est imat ion of the system of input demand and cap i t a l accumulation equations fo r three a l t e rna t i ve models, the "genera l " expectat ions model, the adaptive expectat ions model, and, fo r comparison, the s t a t i c expectat ions model, a l l f o r annual U.S. manufacturing data 1948-77. The data used are discussed in Appendix C. Because of the lags and leads necessary to construct the data required f o r the model, the e f f e c t i v e part of t h i s sample i s 1949-76. The major r e su l t s considered here are based on the f u l l 1949-76 sample, although for comparison the model was a lso estimated fo r the 1949-71 time per iod . Due to the mu l t ip le appearance of x in these formulas, i t was d i f f i c u l t to obtain convergence with many f ree parameters with in the x formula. It was therefore again necessary to cons t ra in some a p r i o r i ; y^ and y^ were constra ined to zero analogous to the monopoly model in Essay 1. It was not poss ib le to determine the v a l i d i t y of these cons t ra in ts fo r when the cons t ra in ts were re laxed, the term within the three corresponding square root s igns became negat ive, causing the est imat ion procedure to terminate. Given the imp l i ca t ions of the monopoly framework and the s t a t i c model, however, these cons t ra in ts appeared the most j u s t i f i a b l e . In add i t i on , in some cases when y^ was allowed to be a free parameter, the same problem occurred; Y k k and the term wi th in the square root became negat ive. It therefore was a l so necessary in some cases to cons t ra in Y k k to .05, a value cons i s ten t with many r esu l t s I was able to obtain and a lso with the cons t ra in t imposed in Essay 1. Even with these cons t ra in ts imposed the model i s complex to est imate. There are 30 parameters, and some have a p r i o r i s ign const ra in ts imposed by economic theory. These inc lude : YFF .YMM^ »"TV£>0. -150-The cons t ra in ts on Y e e and Y m m imply negative short run own e l a s t i c i t i e s , and on Y ^ and Y ^ r e s u l t in a pos i t i v e adjustment c o e f f i c i e n t and negative cap i t a l own e l a s t i c i t y . Note that in BFW (1979) i t was a lso necessary that Y £ Y » Y M Y > 0 ' a n d Y Y K < 0 ' w n 1 c n w e r e required for pos i t i v e output e l a s t i c i t i e s . However, with the funct iona l form spec i f i ed here where Y appears in many more places than in the usual quadratic form, these r e s t r i c t i o n s have inconc lus ive imp l i ca t ions . In add i t i on , cons iderat ion of the expectat ions c o e f f i c i e n t s from the general model above suggests i n t u i t i v e l y that u Y<0, and u E, (>) 0 as E and M are subst i tu tes and complements, r e spec t i ve l y . Keeping these r e s t r i c t i o n s in mind, I now consider the parameter estimates from var ious expectat ions spec i f i c a t i ons reported in Table II-1 fo r the 1949-76 time per iod and Table II-2 for the 1949-71 time per iod . F i r s t cons ider the reported R ' s , which again are ca l cu la ted as the simple c o r r e l a t i o n c o e f f i c i e n t between actual and f i t t e d values squared. Th is i s of 2 course one i nd i ca t i on of goodness of f i t . The R f igures reported in Table 2 1-2 therefore are the R from th i s supplementary c a l c u l a t i o n . From these c o e f f i c i e n t s i t appears that there i s l i t t l e to choose among the models on the bas is of the va r i ab le input demand equat ions; the investment equation " f i t " d i f f e r e d subs tan t i a l l y across s p e c i f i c a t i o n s . The most general model exh ib i ted a much bet ter co re l a t i on c o e f f i c i e n t (.6228) than the adaptive expectat ions model (.3260) which in turn behaves much bet ter than the s t a t i c model ( .1981). Th is impl ies that the more general expectat ions models " f i t " the data be t te r , but s ince they a lso inc lude many more free parameters in the K equat ion, the imp l i ca t ions are not at a l l conc lus i ve . 2 Note a lso the R 's reported fo r the 1949-71 subsample in Table I1-2. In t h i s case both the s t a t i c and adaptive expectat ions spec i f i c a t i ons improve subs tan t i a l l y as compared to the f u l l sample. Th is i s to be expected, -151-TABLE II-l Parameter Estimates fo r A l t e rna t i ve Expectat ions S p e c i f i c a t i o n s , 1949-76 ( t - s t a t i s t i e s in parentheses) S t a t i c Expectat ions .499 (14.42) .003 (1.27) -.005 (-2.16) -.026 (-1.68) .05 3.781 (5.37) -.585 (-1.24) -.127 (-3.60) .008 (1.34) -.004 (-3.33) .004 (2.15) .031 (4.00) .0007 (3.76) Adaptive Expectat ions .505 (4.745) .005 (.636) -.005 (-2.306) -.023 (-1.439) .314 (2.308) 7.564 (3.989) -1.580 (-.843) -.191 (-1.369) .006 (1.044) -8.886 (-3.567) -.094 (.877) -.00005 (-.232) -.008 .029 (4.234) .0005 (3.148) General Expectat ions .336 (3.814) -.011 (-1.700) -.005 (-2.389) -.027 (-1.730) .081 (.715) 4.344 (2.425) 1.344 (.863) .090 (.771) .0065 (1.208) -5.864 (-2.584) .098 (.955) .0003 (1.854) -.008 (.030) .030 (4.328) .0005 (2.940) -152-TABLE II-l (cont 'd) Parameter Est imates, A l t e rna t i ve Expectations S p e c i f i c a t i o n s , 1949-76 S ta t i c Expectat ions Adaptive Expectat ions General Expectat ions Y EK YEK Y Ey a Mt YMk YMK «M YMy "y "E "M "K i "E .219 .302 .269 (3.76) (4.829) (4.389) -.265 -.337 (-1.213) (-1.645) -.00003 -.00002 -.00002 (-4.44) (-3.793) (-3.550) .0002 -.003 -.004 (.12) (-2.526) (-3.596) -1.355 .476 .003 (-5.98) (.913) (.007) -3.648 -5.335 (-1.791) (-3.207) .716 .583 .617 (25.72) (13.762) (15.732) -.00009 .00008 .0001 (-1.21) (1.282) (1.966) .001 (2.488) -2.776 (-4.583) 3.441 (4.036) -.788 (-1.892) 2.523 (1.426) -7.141 (-3.603) -153-TABLE II-l (cont 'd) Parameter Est imates, A l t e rna t i ve Expectations S p e c i f i c a t i o n s , 1949-76 S t a t i c Expectations Adaptive Expectat ions General Expectat ions -.0095 (-3.732) R2's L .9908 .9866 .9935 E .9910 .9914 .9910 m .9927 .9959 .9953 k .1981 .3260 .6228 |A | (1976) .083 .170 .104 -154-TABLE II-2 Parameter Est imates, A l t e rna t i ve Expectations S p e c i f i c a t i o n s , 1949-71 ( t - s t a t i s t i c s in parentheses) S t a t i c Expectations .292 (5.253) -.0082 (-1.874) -.0064 (-1.299) -.0188 (.969) .05 1.304 (2.375) 2.346 (2.612) .054 (.697) .0082 (1.445) .0003 (1.323) -.0072 (-1.893) Adaptive Expectations .308 (2.343) -.005 (-.583) -.015 (-3.148) -.030 (-1.432) .079 (.538) 3.916 (2.194) 1.770 (.807) -.0095 (-.066) .008 (1.467) -9.432 (-4.561) .066 (.448) .0003 (1.247) -.008 General Expectations .300 (9.599) -.008 (-3.356) -.014 (-2.811) -.025 (-1.070) .05 2.461 (1.455) 1.856 (3.084) .040 (.929) .0075 (1.341) -6.884 (-3.294) -.014 (-.084) .0004 (9.219) -.008 .035 (3.666) .043 (4.434) .0325 (3.502) -155-TABLE II-2 (cont 'd) Parameter Est imates, A l t e rna t i ve Expectations S p e c i f i c a t i o n s , 1949-71 S t a t i c Expectat ions Adaptive Expectations General Expectat ions a E t YEK YEK a Mt a M YMy u y "E "M "K i .0008 (2.900) .225 (4.340) -.00004 (-5.559) .0001 (.082) -1.786 (-7.229) .774 (21.030) -.0002 (-2.238) .0003 (1.295) .326 (5.845) .070 (.319) -.00003 (-4.497) -.0026 (-1.755) -.689 (-1.907) -3.768 (-1.845) .693 (15.639) -.00001 (-.139) .0004 (1.876) .482 (8.683) .551 (2.752) -.00003 (-5.145) -.002 (-1.577) 1.561 (3.109) 1.547 (.910) .524 (10.052) .00002 (.392) .003 (.241) 1.986 (.536) .740 (.195) -1.505 (-2.703) 2.170 (.638) -156-TABLE 11-2 (cont 'd) Parameter Est imates, A l t e rna t i ve Expectations S p e c i f i c a t i o n s , 1949-71 S t a t i c Expectat ions Adaptive Expectat ions General Expectat ions urn * 186 (.666) uv -.008 (2.547) R2's L .9923 .9840 .9841 E .9909 .9934 .9941 M .9897 .9950 .9962 K .5426 .5078 .6183 x | (1971) .170 .117 .117 -157-however, fo r the 1947-71 per iod was one of much less va r i a t i on and i s therefore more e a s i l y charac ter ized by simple expectat ions assumptions, even s t a t i c expectat ions . In add i t i on , the |x| estimates reported in these tables seem quite sma l l , ranging from .083 fo r the s t a t i c expectat ions model to .170 fo r the adaptive expectat ions model fo r the 1949-76 time per iod . Reca l l , however, the b r i e f d iscuss ion above on the in te rp re ta t i on of x. Rather than r e f l e c t i n g a simple proport ion between the des i red long run cap i t a l stock K** and the actual l e v e l , i t charac te r izes both the proport ion of the d i f f e rence between K and K* and depends on the adjustment process because i t i s embedded in the weighted average of future v a r i ab l e s . It therefore i s d i f f i c u l t to i n t e rp re t w e l l , although the .170 c o e f f i c i e n t remains i n t u i t i v e l y more p l aus ib l e than .083, which suggests extremely slow adjustment. The parameter estimates fo r the models appear to accord well with economic theory. The adaptive and general expectat ions spec i f i c a t i ons s a t i s f y a l l the r e s t r i c t i o n s l i s t e d above fo r both time per iods , although for the 1949-71 general expectat ions model the .05 r e s t r i c t i o n on had to be imposed. This suggests that the more complex model might not charac ter ize periods well that conta in l i t t l e v a r i a t i o n ; perhaps in th i s case i t i s d i f f i c u l t to i den t i f y the extra parameters of the expectations process. Note a l so that the one i n t u i t i v e r e s t r i c t i o n on the "composite c o e f f i c i e n t s " fo r the general model, that of uy» i s s a t i s f i e d . In add i t i on , and ufa are pos i t i v e and negat ive, r e spec t i ve l y , poss ib ly i nd i ca t ing a complementary r e l a t i onsh ip between energy and c a p i t a l , and s u b s t i t u t a b i l i t y between mater ia ls and c a p i t a l . This i s , however, somewhat speculat ive s ince these e l a s t i c i t i e s are f a i r l y complex. The r e su l t s fo r the s t a t i c expectat ions case exh ib i t the same problem but in reverse . Although in the 1949-71 s p e c i f i c a t i o n the r e s t r i c t i o n s a l l -158-ho ld , once the years of l a rger va r i a t ions are added y^ K could not be estimated with in the model. In add i t i on , Y Y|< becomes p o s i t i v e , whereas Y^y a n d Y£Y take o n negative va lues , g i v ing an ind i ca t i on of problems with output e l a s t i c i t i e s . As mentioned above, in t h i s case such a r e su l t i s not at a l l conc lus ive s ince the funct iona l form i s a complex funct ion of Y. It does seem to i nd i c a t e , however, that although the s t a t i c expectat ions model f i t s the data fo r stable time periods r e l a t i v e l y we l l , i t i s not as able to charac te r ize periods of large f l u c tua t i ons . These parameter estimate d iscuss ions have been s t r i c t l y in terms of s t ruc tura l est imates. In add i t i on , c o e f f i c i e n t s of the expectat ions process were der ived for the IMA(1,1) case to model adaptive expectat ions , and for a few s l i g h t l y more general models to determine whether the one period se r i a l c o r r e l a t i o n assumption was j u s t i f i e d . Since the expectat ions process i s in terms of r e l a t i v e p r i c e s , the most important var iab le for modeling the mechanics of the expectat ions process would appear to be output. However, ARIMA regress ions on the output va r iab le show l i t t l e evidence of any s tochast i c s t ruc tu re , i nc lud ing ind i ca t i ons of adaptive expectat ions . Further information on the r esu l t s of the ARIMA est imation are provided in Appendix D. The s i gn i f i c ance of the expectat ions process fo r modeling the f i rm ' s behavior i s , however, a much more complex question than simple ARIMA processes could r e f l e c t . The s ign i f i c ance of incorporat ing the non-stat ic expectat ions process can be pursued fur ther and addressed more d i r e c t l y in terms of the e l a s t i c i t y estimates reported in Table I1-3. Note f i r s t that the myopic long run (MLR) and long run (LR) estimates are the same by d e f i n i t i o n in the s t a t i c expectat ions case , as K*=K**. In the non-stat ic expectat ions case, however, the va r i a t i on between the MLR and LR e l a s t i c i t i e s i so l a t e numerical ly the d i f f e rence between adjustment toward a long run cap i ta l stock def ined in - 1 5 9 -TABLE I1-3 Pr ice and Output E l a s t i c i t y Est imates, A l t e rna t i ve Expectat ions Spec i f i c a t i ons 1 9 4 9 - 7 6 (Reported fo r 1 9 7 1 ) S ta t i c Expectations Adaptive Expectat ions General Expectations SR LR SR MLR LR SR MLR LR e L L - . 0 2 4 - . 0 4 1 - . 0 6 1 - . 1 0 9 - . 1 8 3 - . 0 7 0 - . 1 7 4 - . 1 6 0 e L E - . 0 1 0 - . 0 1 9 - . 0 0 2 - . 0 0 7 - . 0 0 5 - . 0 0 4 . 0 1 8 . 0 0 6 e L M . 0 3 4 . 0 9 7 . 0 6 3 . 0 7 4 . 0 7 1 . 0 7 4 . 0 7 4 . 0 7 7 e L K 0 - . 0 3 8 0 . 0 2 8 . 0 2 2 0 . 0 8 2 . 0 3 8 G E L - - 6 3 2 - . 1 2 6 - . 0 1 0 . 0 5 1 . 1 8 3 - . 0 2 6 . 1 5 8 . 2 8 7 < - E E - . 1 2 6 - . 1 6 9 - . 1 2 0 - . 1 4 0 - . 1 3 8 - . 1 2 6 - . 2 1 5 - . 1 9 4 e E M . 1 8 9 . 4 3 7 . 1 2 9 . 1 2 4 . 1 2 2 . 1 5 2 . 2 1 0 . 1 8 9 e E K 0 - . 1 4 2 0 - . 0 3 5 - . 0 3 4 0 - . 1 5 2 - . 1 3 7 e M L . 0 1 7 . 0 4 3 . 0 3 0 . 0 3 7 . 0 5 2 . 0 3 6 . 0 3 5 . 0 3 6 e M E . 0 1 4 . 0 2 9 . 0 1 0 . 0 0 9 . 0 0 9 - . 0 1 2 . 0 1 1 . 0 1 1 e M M - . 0 3 1 - . 1 4 0 - . 0 4 0 - . 0 4 2 - . 0 4 2 - . 0 4 7 - . 0 4 7 - . 0 4 7 E M K 0 . 0 6 9 0 - . 0 0 4 - . 0 0 4 0 - . 0 0 0 0 9 - . 0 0 0 0 9 e K L 0 - . 2 4 6 0 . 1 7 4 . 5 3 6 0 1 . 7 0 0 1 . 5 9 1 e K E 0 - . 1 3 6 0 - . 0 3 0 - . 0 2 8 0 - . 3 5 9 - . 1 8 4 -160-Pr ice and Output E l a s t i c i t y Est imates, A l t e rna t i ve TABLE II-3 (cont 'd) Expectations Spec i f i c a t i ons 1949-76 S t a t i c Expectat ions Adaptive Expectations General Expectations SR LR SR MLR LR SR MLR LR 0 1.004 0 -.047 -.045 0 -.004 -.002 0 -.621 0 -.098 -.094 0 -1.336 -.684 .692 .406 .893 .927 .937 .717 1.165 1.216 .193 -.222 .200 .411 .405 .278 1.239 1.117 1.003 1.165 1.050 1.079 1.077 1.137 1.132 1.132 0 1.901 0 .551 .528 0 7.504 3.841 -161-terms of cur rent va r iab les or in terms of the present value of a stream of non-constant exogenous va r i ab l e s . The d i f f e rence between the MLR and LR response, however, as represented by both the adaptive expectat ions and general expectations r e s u l t s , are not extremely l a rge . For example, take the case of energy. It appears that in the MLR, energy i s f a i r l y subst i tu tab le with both mater ia ls and l abor , although i t exh ib i t s a f a i r l y low pr i ce response to f l uc tua t ions in i t s own p r i c e . In the long run, the c ross-e f f e c t with mater ia ls i s approximately the same, but energy becomes inc reas ing ly subst i tu tab le for labor and s l i g h t l y less responsive to i t s own p r i c e . This appears to ind ica te that with an increase in the p r i ce of l abor , given that nothing e lse w i l l change in the fu tu re , labor and energy are complements. However, a l lowing fo r future changes the f i rm assumes that e i the r th i s p r i ce increase or other e f f e c t s w i l l a r i se that w i l l cause energy to be even more subst i tu tab le fo r l abor . In con t r a s t , i n terms of i t s own e l a s t i c i t y , energy w i l l be f a i r l y i n e l a s t i c in response to a change in i t s own pr i ce given no change in any other var iab les on the path to the long run, but i f other va r i ab les are changing the energy pr ice response w i l l be even s l i g h t l y l ess e l a s t i c because the f i rm assumes the p r i ce of energy w i l l not remain as high r e l a t i v e to other p r i c e s . Note that ne i ther one of these responses i s as la rge as one would think i s i n t u i t i v e l y p l a u s i b l e , although the r e su l t i s quite robust across s p e c i f i c a t i o n s . I should a lso note that 1971 happens to be the year of the second smal lest estimated energy e l a s t i c i t i e s over the ent i re time pe r iod , second only to 1970. Since the c o e f f i c i e n t s fo r these e l a s t i c i t i e s are constant but the data var ies over t ime, va r i a t ions in a l l e l a s t i c i t i e s over the sample per iod w i l l be found. Over the en t i r e sample pe r iod , f o r example, the average MLR own p r i ce e l a s t i c i t y f o r energy i s approximately - .22 , which i s s t i l l low but not qui te as low. Thus, although the year 1971 was chosen as i t was the -162-end of the sample used in previous work and therefore allows for comparisons, i t does not appear to be p a r t i c u l a r l y representat ive fo r the energy e l a s t i c i t y . Most other e l a s t i c i t i e s are more robust over the sample pe r iod . Other p r i ce e l a s t i c i t i e s can be given s im i l a r in te rpre ta t ions to the energy e l a s t i c i t i e s . By con t ras t , however, the output e l a s t i c i t i e s have s l i g h t l y d i f f e r e n t i n t e rp r e t a t i ons . The question in th i s case i s , when output changes, do inputs change in proport ion to keep the input mix constant . I f they do, there i s constant returns to sca le (CRTS) with a l l long run output e l a s t i c i t i e s equal to un i t y . In th i s case, i t appears that they do not, although the r e su l t s vary fa r more between spec i f i c a t i ons than between MLR and LR, and more than for the pr i ce e l a s t i c i t i e s . Focussing on the adaptive expectat ions case for now, I note that i t appears that both cap i t a l and energy increase less than propor t ionate ly in the long run, implying increas ing p roduc t i v i t y of these inputs in the long run, while the reverse i s true for ma te r i a l s . A l l these r esu l t s vary l i t t l e between MLR and LR. S p e c i f i c a l l y , with an output change and no other e f f e c t , labor adjusts almost p ropor t iona te l y . Once other adjustments have been made on the path to the long run, however, labor adjusts s l i g h t l y more. By con t ras t , energy increases by only about 40 percent , given a one-shot output change, i nd i c a t i ng strong inc reas ing returns to energy, and drops s l i g h t l y again once other f l uc tua t ions are accommodated. The d i f f e rence i s i n s i g n i f i c a n t , however. In sp i t e of t h i s non-homotheticity, overa l l returns to sca le appear to be f a i r l y constant ; returns to scale are measured as approximately 1.02 in both the MLR and LR. Thus, overa l l in terms of the d i f f e rence between MLR and LR e f f e c t s , the incorporat ion of non-stat ic expectat ions does not appear to make very much d i f f e r e n c e . Comparison between the var ious models, however, could s t i l l ind ica te an important e f f e c t of the expectat ions process, as with non-stat ic expectat ions the f i r m ' s behavior changes over the en t i re time path inc lud ing -163-the current pe r i od . Thus i t i s useful to compare the models with respect to two general c r i t e r i a , f i r s t whether the overa l l pattern of response i s very d i f f e r e n t with the d i f f e r e n t pattern of expectat ions , and, second, whether the short run e f f e c t i s very d i f f e r e n t . To pursue these quest ions, I compare the e l a s t i c i t y estimates for the various expectat ions spec i f i c a t i ons in Table I1-3. The overa l l pattern appears to d i f f e r much more between the s t a t i c and non-stat ic expectat ions spec i f i c a t i ons than between the two expectat ions s p e c i f i c a t i o n s , except fo r the returns to s ca l e . S p e c i f i c a l l y , in terms of p r i ce e l a s t i c i t i e s , the short run impacts represented by the s t a t i c expectat ions as compared to the non-stat ic expectat ions formulat ions are s im i l a r in s i gn , although not necessar i l y magnitude, and d i f f e r a f a i r amount in the long run. Although the own-price e l a s t i c i t i e s are s i m i l a r , labor and energy, and labor and c a p i t a l , f o r example, exh ib i t complementarity in the long run fo r the s t a t i c expectat ions model, but s u b s t i t u t a b i l i t y f o r the remaining two models. Mater ia ls and cap i t a l appear to be subst i tu tes in the s t a t i c framework but complements in the non-stat ic framework. These d i f fe rences in the i n t e r- re l a t i onsh ips over time are extremely important fo r p red i c t ing the e f f e c t s of any type of shock on the economy over more than the immediate time per iod . Note a lso that the output e f f e c t s are very d i f f e r e n t across a l l of the models; although the overa l l long run returns to scale in the adaptive expectat ions model i s very c lose to un i t y , returns to sca le in the s t a t i c case appear to be approximately 1.58 and in the general case .81 , i nd i ca t i ng extreme increas ing and decreasing returns to s ca l e , r e spec t i ve l y , ne i ther of which i s a very p l aus ib l e r e s u l t . Hence the adaptive expectat ions model appears to y i e l d the most p l aus ib l e empir ica l r e s u l t s . Part of the problem in the general expectat ions model may be an extremely large cap i t a l output r a t i o , -164-which i s very v o l a t i l e over the sample pe r i od . The large e l a s t i c i t y of demand fo r c ap i t a l with respect to an increase in output ind ica tes strong decreasing returns to c a p i t a l . This r e su l t may suggest a problem separately i den t i f y i ng the expectat ions process and sca le e f f e c t in t h i s unconstrained model, so that imposing CRTS and thus some st ructure on the sca le e f f e c t even i f not on the expectat ions process may r e su l t in more p l aus ib l e r e s u l t s . The overa l l impression therefore appears to be that non-stat ic expectat ions i s important to take in to account, p a r t i c u l a r l y fo r i n f e r r i n g adjustment behavior over t ime, but that imposing an adaptive expectat ions s t ructure on the model does not appear to a f f e c t the estimates subs tan t i a l l y while al lowing fo r more p l aus ib l e and less v o l a t i l e scale e f f e c t s . The adaptive expectat ions framework, in other words, appears to r e f l e c t the important impacts of non-stat ic expectat ions at l ess cos t in terms of overa l l s t ruc tu re , and thus i s , in a sense, a "pre fe rab le " or at l e as t j u s t i f i a b l e empir ica l representat ion of the formation of expectat ions within t h i s type of model. This conc lus ion i s fu r ther supported by the evidence of e l a s t i c i t i e s fo r the general model over the en t i r e time pe r i od ; the e l a s t i c i t y estimates in general are more v o l a t i l e than fo r the adaptive model, p a r t i c u l a r l y fo r the e a r l i e r years in which there was r e l a t i v e l y l i t t l e va r i a t i on to i den t i f y the parameters in the unconstrained s t ruc tu re . Note f i n a l l y tha t , given the ind i ca t ions from the monopoly model in Essay 1, r e su l t s from th i s model may be improved i f gross rather than net investment were assumed to be subject to increas ing costs of adjustment. However, such a s p e c i f i c a t i o n i s more complex, and s ince previous studies were based on net investment, I have reta ined the net investment model here. In future work i t may be useful to combine simultaneously the genera l iza t ions of Essays 1 and 2, p a r t i c u l a r l y fo r the r e l a t i v e l y simple adaptive expective expectat ions model. -165-VI. Concluding Remarks "We may be asking too much of our data . We want them to t e s t our theo r i e s , provide us with estimates of important parameters and d i s c lose to us the exact time form of the i n t e r r e l a t i onsh ips between the var ious va r i ab l e s . Progress in th i s area i s l i k e l y to be slow un t i l we have a much better theore t i ca l base f o r imposing a time-lag s t ructure on the da ta . " (G r i l i ches (1967), p. 18) These concerns expressed by Zvi G r i l i c h e s in 1967 have provided over the l a s t f i f t e e n years the basis fo r much work on modeling the dynamic intertemporal nature of the f i rm ' s decision-making process. Various attempts have been made to develop a theore t i ca l bas is fo r time-lag st ructures based on two c r u c i a l aspects of intertemporal behavior, the existence of stocks and non-constant future time paths of re levant exogenous v a r i ab l e s . In t h i s Essay I have attempted to incorporate both of these components into a t h e o r e t i c a l l y j u s t i f i a b l e model der ived from the determin is t i c framework of a f i r m ' s dynamic opt imizat ion process. The r e su l t i ng model al lows fo r incorporat ion of c r u c i a l aspects of expectat ions formation by imposing s u f f i c i e n t s t ructure to i den t i f y var ious important components of the investment behavior of f i rms , while re ta in ing ana l y t i ca l and empir ica l t r a c t a b i l i t y . This approach therefore i s an attempt to develop a " theore t i ca l base fo r imposing a time lag s t ructure on the data" as G r i l i c h e s a l ludes t o , an attempt which y i e l d s important theore t i ca l and empir ica l evidence on the intertemporal i n t e r r e l a t i onsh ips among var iab les determined by the behavior of the f i r m . T h e o r e t i c a l l y , I f i n d that i t i s poss ib le to develop a s t ruc tura l model imposing both lags from stock r e l a t i onsh ips and from expectat ions format ion, i n terms of e x p l i c i t dynamic opt imizat ion s t ruc tu re . Emp i r i c a l l y , the major f i nd ing i s that an adaptive expectat ions s p e c i f i c a t i o n appears to represent very n i ce ly the most important aspects of non-stat ic expectations behavior while imposing enough s t ructure to obtain p l aus ib l e estimates of the remaining -166-technologica l s t ruc tu re . This adaptive expectat ions framework, in t u r n , appears bet ter able to charac te r ize the f l uc tua t ions in input demand observed in the 70's than most ex i s t i ng s t ruc tura l models which have tended to re ly on s tochast i c spec i f i c a t i ons and/or e a r l i e r more stable data sets to obtain sens ib le r e s u l t s . More s p e c i f i c a l l y , wi th in t h i s s t ructure there appears to be l i t t l e d i f f e rence between long run subs t i tu t i on impacts in terms of myopic as compared to true long run e f f e c t s . However, incorporat ion of the non-stat ic expectat ions process appears to capture important d i f fe rences between short run and long run impacts. As compared to the more general case , the adaptive expectat ions case appears less v o l a t i l e , p a r t i c u l a r l y with respect to the evidence of sca le e f f e c t s , while s t i l l cha rac te r i z ing remaining impacts p l aus ib l y . Thus t h i s Essay provides useful and suggestive empir ica l content that supports the procedures used, but a lso contains a more major methodological message. S p e c i f i c a l l y , what I have attempted to accomplish in t h i s Essay has been to incorporate non-stat ic expectat ions d i r e c t l y in to the s t ructure of the determin is t i c model, rather than to r e l y on complex stochast ic spec i f i c a t i ons to incorporate the e f f e c t s of expectat ions . Although th i s requires ce r t a i n r e s t r i c t i o n s , the v a l i d i t y of the r e s t r i c t i o n s can be emp i r i ca l l y i n f e r r e d , and seem empi r i ca l l y to be substant ia ted . In add i t i on , within t h i s framework i t i s poss ib le to compare resu l t s from models with a simple s p e c i f i c or unres t r i c ted expectat ions s t ruc tu re . In my empir ica l example, t h i s c a p a b i l i t y r e su l t s in the evidence discussed above that an adaptive expectat ions s t ructure can be j u s t i f i e d emp i r i c a l l y . F i n a l l y , as well as a l lowing useful theore t i ca l and empir ica l inferences to be made, incorporat ing non-stat ic expectat ions e x p l i c i t l y in to the s t ruc tura l model increases the r ichness of the dynamic model in terms of -167-app l i ca t ions to other l i t e r a t u r e . For example, a l lowing e x p l i c i t l y f o r expectat ions formation f a c i l i t a t e s the l inkage of t h i s type of model with investment demand models based on Tob in ' s q. Within my framework Tob in ' s q can be der ived in terms of the shadow value of c a p i t a l , with future expectat ions and thus true present value opt imizat ion taken e x p l i c i t l y in to account. Th is i s an important aspect of the q concept, s ince market va luat ion of a f i rm depends on expectat ions of the future value of current c ap i t a l s tocks . Dynamic measures a l so can be used to generate measures of capac i ty u t i l i z a t i o n . Expectat ions are important in t h i s context s ince current investment and capac i ty are based on expectat ions of future output demand. This idea i s pursued fu r ther in the next Essay. -168-FOOTNOTES 1 This i s a p a r t i c u l a r l y important genera l iza t ion for output. The BFW s p e c i f i c a t i o n i s based on a normalized cos t funct ion so that a s t a t i c p r i ce expectat ions assumption r e f l e c t s myopic expectat ions with respect to r e l a t i v e p r i c e s , which i s not as r e s t r i c t i v e as i f i t were with respect to absolute p r i c e s . However, the assumption that the f i rm expects the, current output leve l to continue in to the fu tu re , given observed substant ia l h i s t o r i c a l va r i a t ions in output demand, i s not an appeal ing s i m p l i f i c a t i o n . 2 Nerlove (1972), p. 221, quoting H. Schul tz (1938). 3 Although they were o r i g i n a l l y der ived in order to give theore t i ca l foundations to d i s t r i bu t ed lag models. 4 "The s t a t i c concept of long run equ i l ib r ium retarded the development of econometr ica l ly re levant dynamic micro theory. This i s not to say that comparative s t a t i c s has no place in economic theory, but merely that i t i s not a useful approach when one i s e x p l i c i t l y concerned with the dynamics of behavior" (Nerlove et al (1979), p. 293). 5 See Kennan (1979) and Arrow (1978). 6 Meese (1980). 7 Or perhaps more re levant l y to no targets at a l l but to changing expectat ions o f the p r o b a b i l i t y d i s t r i b u t i o n s o f future p r i c e s . 8 Nerlove (1972) p 232. 9 Nickel 1 (1978), Ch. 3. 1 0 See Nagatani (1978). 1 1 Arrow (1978). 1 2 Arrow, p. 159. Note that t h i s i s e spec i a l l y important i f the agent i s a r i sk-aver te r so that h is dec is ions take into account moments other than the mean (expected va lue ) . 13 Arrow (1978), p. 160. Arrow pursues th i s fur ther in the context that the economic world i s complex and va r i ed , and each ind i v idua l agent only has l im i t ed in format ion: "I disagree with the widely accepted propos i t ion that econometric models should have expectat ions cons i s ten t with them. To the extent , i t i s argued, that the economic theory underly ing the model involves a n t i c i p a t i o n s , the an t i c i pa t i ons that appear in the model as determining ind iv idua l behavior should be equal to the forecasts made from the model. More genera l ly in f a c t , I would disagree with the weaker p ropos i t i on , that an t i c i pa t i ons made by i nd i v i dua l s should be necessar i l y dependent upon broadly ava i l ab le general data about the economy and in pa r t i cu l a r about government ac t i ons . - 1 6 9 -Let me take up the f i r s t p ropos i t i on . It i s the essence of the decent ra l ized economy that i nd i v i dua l s have d i f f e r en t informat ion. Each ind i v idua l i s spec i a l i z ed i n c e r t a i n a c t i v i t i e s and has i n general spec i a l i z ed knowledge about those a c t i v i t i e s . There i s no reason therefore why h is forecasts should be based only on the rather general kind of information which the econometrician can use. . . . In shor t , one object ion to what may be c a l l e d the cons i s ten t expectat ions hypothesis (the hypothesis that the an t i c ipa t ions imbedded in any model should be the best fo recas t from that model) i s that t h i s i s very fa r from a set of ra t iona l expectat ions fo r any given agent. Each agent ought r a t i o n a l l y to base h is an t i c i pa t i ons on a l l the information at h is disposal and th i s may include a great many fac ts and observations not ava i l ab le to o thers . . . . Thus the an t i c i pa t i ons of the d i f f e r e n t economic agents are not only not based on the same general economic model but they should i n general d i f f e r considerably from each other . . . I a l so want to argue that they w i l l not necessar i l y use a l l the information contained in an econometric model. In f a c t , the two propos i t ions are in t imate ly l inked though they seem to move in opposite d i r e c t i o n s . We have to assume that information-processing a b i l i t y i s s ca rce . " (pp. 1 6 4 - 1 6 5 ) 1 4 B . Friedman, p. 2 4 . . 1 5 See Friedman ( 1 9 7 9 ) and Frydman ( 1 9 8 1 ) . 1 6 The foundation o f the ana lys i s i n Friedman's study i s the "confus ion surrounding the meaning of ' r a t i o n a l ' expectat ions due to a f a i l u r e to d i s t i ngu i sh between (a) the general assumption that economic agents use e f f i c i e n t l y whatever information i s a v a i l a b l e , and (b) a spec i f i c assumption i den t i f y i ng the ava i l ab le set of informat ion" (p. 2 5 ) . The information assumption i s reasonable, in the s t r i c t economic sense that people use informat ion un t i l i t s marginal product equals i t s marginal c o s t , which may be zero . The information a v a i l a b i l i t y assumption i s more quest ionable . It i s based on the a b i l i t y of economic agents to form expectat ions charac te r ized by cond i t iona l subject ive d i s t r i b u t i o n s of outcomes that are equiva lent to the cond i t iona l ob ject ive d i s t r i b u t i o n s of outcomes given by the re levant economic theory (Friedman, p. 2 5 ) . 1 7 Note that p a r t i a l r a t i o n a l i t y in which the ant i c ipa ted ser ies i s a funct ion only of the past time ser ies of i t s own va lues , or adaptive expectat ions are spec ia l cases of the " r a t i o n a l i t y " hypothesis and thus can be tes ted . 1 8 Adjustment c o s t s , and therefore f ixed stocks and " pa r t i a l adjustment" could a lso induce a time lag of adjustment, but as long as expectat ions were r a t i o n a l , these components of the economic s t ructure would be taken in to account so that the f i rm would always be at an "optimum" po in t . 1 9 Nerlove et al ( 1 9 7 9 ) , p. 2 9 3 . 2 0 An example might be an e l e c t r i c u t i l i t y . 2 1 I.e., "Had we assumed separable adjustment cos t s , investment would not have entered the determination of the marginal product of l abor , and we cou ld have used (the opt ima l i t y cond i t ion fo r labor) to reduce our problem to the optimal determination of input l eve l s over time of a s ing le f ac to r of -170-product ion, c a p i t a l . Nonseparable adjustment costs require the optimal cur rent l eve l s of gross investment and labor input to be j o i n t l y determined" (Nerlove (1972), p. 240). 2 2 i am indebted to Stephen Nickel 1 fo r h is help i n pr iva te correspondence to extend his d i s c re te model (1977) to an analogous continuous case . 2 3 See Nickel 1 (1977), pp. 4-5. 2 4 See Kokkelenberg (1979). 2 5 Nerlove (1972), p. 239. 2 6 Ner love, p. 239. 2 7 Malinvaud (1969) shows that ce r ta in t y equivalence may be a useful and "c lose enough" approximation even i f these r e s t r i c t i o n s do not ho ld : "The f i r s t order ce r t a in t y equivalence property impl ies that the optimal i n i t i a l dec i s ion changes l i t t l e with the degree of uncerta inty as long as the l a t t e r i s sma l l . Dec is ions taken on the bas is of models in which the random disturbances are neglected should be c lose to optimal as long as these disturbances have zero expected values and the d i f f e r e n t i a b i l i t y condi t ions are s a t i s f i e d . " (Malinvaud, p. 715)) In add i t ion one may be able to to t e s t the "c loseness" of the approximation by imposing the cond i t i ons , e . g . , l i n e a r i t y r e s t r i c t o n s , on the dec is ion func t ions , and tes t f o r t h i s more r e s t r i c t i v e s p e c i f i c a t i o n . There are problems with th i s concept, however, fo r the a l t e rna t i v e hypothesis i s not well de f ined . Thus the t e s t i s not powerful ; i t i s not d i s c r im ina t i ng , and thus not very u se fu l . 2 8 Note that the concept of " r a t i o n a l i t y " in th i s case i s s l i g h t l y d i f f e r e n t than that used fo r the REH above. 2 9 Note that in general the disturbance terms from these equations w i l l be s e r i a l l y co r r e l a t ed as in the simple adaptive expectat ions model spec i f i ed above. 30 Nerlove et a l , p. 294. 31 E . g . , ,as i t imposes a c e r t a i n s t ructure on the " r a t i o n a l " d i s t r i bu ted lag discussed above. 3 2 " I f , in f a c t , there i s a lag in response to pr i ce changes in the formation of expectat ions about future p r i c e s , th i s suggests a weight s t ructure in which the weights increase at f i r s t and a f t e r some point decrease" (Solow (I960)). 33 Th is i s true in the sense that t h i s i s a pa r t i a l model so ra t iona l expectat ions requires only s p e c i f i c a t i o n of expectations by past values of the same exogenous va r i ab le and not the parameters of the s t ructura l model. 3 4 For examples o f s i m i l a r methods, see Sargent (1979) fo r the former and Abel (1979) for the l a t t e r approach. 3 ^ This i s not a problem i f we charac te r ize the f i rm ' s problem as one of p r o f i t maximization instead of cos t min imizat ion . Since th i s i s an appeal ing -171-property the assumption of p r o f i t maximization would be useful to pursue in future work. 3 6 See Solow (1960) for example. 3 7 Abel (1978). 3 8 This can be assumed given the j u s t i f i c a t i o n s discussed above, or tested for as a r e s t r i c t i o n of the general ARIMA (p,q) s p e c i f i c a t i o n . 3 9 Abel (1978). Analogous ca l cu l a t i ons can be made e x p l i c i t l y for the model considered here, but s ince ir+, i s a vector the expos i t ion becomes complex and provides no more useful informaton. Therefore I have not included t h i s de r i v a t i on . 40 See M. Friedman's d iscuss ion of permanent and t r ans i to r y income. 41 Note that bypassing the c a l cu l a t i on of the expectat ions process for the Koyck transform can be in terpre ted as bypassing the problem of m i s spec i f i c a t i on of expectat ions formation. Thus the general Koyck transform procedure i s preferab le in terms of leav ing l ess p o s s i b i l i t i e s fo r m i s spec i f i c a t i on e r ro r in the f i na l formulat ion; the est imat ion i s not dependent on modeling the expectat ions process c o r r e c t l y . 4 2 This approach a lso suggests a more complicated expectat ions formation process e x p l i c i t l y incorporat ing equation (2 .4 .17) , which in the general case expresses Rt+i i n terms of the innovations in each per iod . Such a procedure has been suggested by F l av in (1977), but i t i s subs tan t i a l l y more complex than the simple form der ived with the adaptive expectat ions s p e c i f i c a t i o n . Thus i f adaptive expectat ions can be assumed the est imat ion procedure i s s i m p l i f i e d . 4 3 I f a s tochast i c s p e c i f i c a t i o n i s imposed on the o r ig ina l equation before the Koyck transform, however, an add i t iona l problem resu l t s in the form of a moving average e r ro r . Then the disturbance term i s co r re l a ted with A K ^ + I whether or not the R t + i term i s a problem. In that case instrumental va r i ab les are again requ i red , which could be future var iab les such as Note that tr-t i s a vector rather than a sca la r as in Abel (1978), so a wide choice of instruments i s a v a i l ab l e . 44 Abel (1978). 45 See Berndt-Fuss-Waverman or Morrison-Berndt 4 6 See Morrison (1981). -172-Essay 3. On the Economic In terpretat ion and Measurement of Optimal Capacity U t i l i z a t i o n I. Introduct ion The 1970's and ear ly 80's have been marked by substant ia l va r i a t i on in trends for nat ional economic growth and labor p roduc t i v i t y . One c ruc i a l development during th i s time with widely ranging impacts has been the dramatic change in energy pr i ce t rends . Although considerable controversy s t i l l ex i s t s concerning the nature of i t s impact, t h i s phenomenon appears to have a f fec ted p roduc t i v i t y and growth through short run changes in output and in the pattern of subs t i tu t ion between cap i t a l and labor . The long run impact i s s t i l l u n c e r t a i n . 1 Values o f Tob in ' s q (the r a t i o of the market value of the f irm to the replacement value o f i t s physica l c ap i t a l ) have a lso dropped sharply s ince 1973, r e f l e c t i n g the " s lugg i sh pace of business investment during the 2 br isk economic expansion which fol lowed the 1974-75 r e cess i on . " The cap i t a l formation slowdown has in turn been l inked to energy pr i ce increases and labor p roduc t i v i t y decreases through the impacts of energy pr i ces on labor-cap i ta l subs t i t u t i on . C l ea r l y these a l t e rna t i ve cha rac te r i za t ions of economic behavior over t h i s time per iod are c l o se l y in ter twined. Such i n t e r r e l a t ed trends h igh l i gh t the importance in economic research o f modeling the intertemporal behavior of the f i rm within an integrated framework. A great deal of research has appeared s t ress ing i n t e r r e l a t i onsh ips in economic i nd i ca to r s such as c y c l i c a l p roduc t i v i t y , capac i ty u t i l i z a t i o n , 3 investment determinants, and stocks of cap i t a l and l abor . Even so, s tudies which have attempted to charac te r ize the f l e x i b i l i t y of the f i rm ' s response to c y c l i c a l v a r i a t i on have t y p i c a l l y been s t a t i c or non-optimizing models, or e l se have depended l a rge l y on s tochast i c processes. Thus, most work to date -173-has not r e f l e c t e d t h e o r e t i c a l l y or emp i r i ca l l y the i n t e r r e l a t i onsh ips among the f i r m ' s dec is ion processes and observed c y c l i c a l economic behavior . Th is has resu l ted in use o f var ious c y c l i c a l i nd i ca to r s which provide l i t t l e i n t e rp re t i v e in format ion, s ince there i s l i t t l e in the way of economic "underpinnings" supporting t h e i r de r i v a t i on . Perhaps the most common measure used today to i n f e r imp l i ca t ions about c y c l i c a l va r i a t ions i s the rate of capac i ty u t i l i z a t i o n . Several measures of capac i ty u t i l i z a t i o n {hereafter, CU) e x i s t , mostly based on peak-to-peak i n t e rpo l a t i on or survey informat ion. Two examples are the Federal Reserve Board and Wharton Measures. These ind ices are reproduced in Table 3.1, and trace annual va r i a t i on s ince 1954. As i s evident from the f i gu res , peaks and troughs of these CU measures, they correspond r e l a t i v e l y c l o se l y to other t yp i ca l c y c l i c a l i nd i ca to r s such as labor p roduc t i v i t y and Tob in ' s q. Note that c y c l i c a l va r i a t ions have been p a r t i c u l a r l y v o l a t i l e s ince the 1973 shocks, and although s t i l l p r o c y c l i c a l , they have not exh ib i ted the strong i n t e r r e l a t i o n s h i p s observed e a r l i e r . For example, labor p roduc t i v i t y measures have not exh ib i ted as strong a p rocyc l i c a l r e l a t i onsh ip with output in the 1970's . The above observat ions ra i se the issue of whether past r e l a t i onsh ips among these ind i ca to rs s t i l l ho ld . I t i s very d i f f i c u l t to answer such a question with t r ad i t i ona l c y c l i c a l measures, s ince for the most part they are not based on an economic s t ruc tu re . As an example, consider the impact of increases in the pr i ce of energy s ince 1973 on capac i ty u t i l i z a t i o n . Such an impact cannot e a s i l y be traced because the mechanisms these measures are based on provide no explanat ion or i n te rp re ta t i on of how changes in the pr i ce of energy a f f e c t the r e su l t i ng est imates. Thus, although major changes in the economy r e su l t i ng from exogenous shocks such as energy p r i ce changes may have caused o ld quant i ta t i ve r e l a t i onsh ips among measured capac i ty u t i l i z a t i o n , -174-investment, labor p roduc t i v i t y , and i n f l a t i o n to change s u b s t a n t i a l l y , the nature of changes cannot be adequately in terpre ted by the current theory. In t h i s essay I demonstrate that movements in ce r t a in c y c l i c a l measures—product iv i ty , Tob in ' s q, and p a r t i c u l a r l y capac i ty u t i l i z a t i o n — a r e not random but can be viewed as the systematic r e su l t of a ra t iona l economic opt imizat ion process undertaken by the f i rm , character ized by the type of dynamic opt imizat ion framework discussed in Essays 1 and 2. Th is e x p l i c i t economic s t ructure allows for theore t i ca l ana lys is of these i n t e r r e l a t i onsh ip s by prov id ing an integrated framework in which to address the questions involved in measuring the f i rm ' s behavior in response to c y c l i c a l shocks. The fo rma l iza t ion of these i n t e r- re l a t i onsh ips emphasizes the importance of economic foundations for these i nd i c a to r s . It provides economic j u s t i f i c a t i o n fo r using these i nd i ca to r s and a lso f a c i l i t a t e s i n te rp re ta t i on of what information i s a c tua l l y conveyed by the numbers. Overal l I f i nd that a general approach can be generated for determining an economic capac i ty u t i l i z a t i o n (CU) measure that i s c l o se l y re la ted to the shadow value of f ixed inputs such as cap i t a l and thus provides useful in te rpre tab le informat ion. Th is i s based on a notion of how capac i ty output i s de f ined , as discussed fo r example by B. Hickman (1964) and others , and given empir ica l content in a r e l a t i v e l y simple framework by Berndt, Morr ison, and Watkins (1981). The impl i ca t ions provided by t h i s procedure have two dimensions. The measure i s useful because i t can be used to in te rp re t and r a t i o n a l i z e the app l i ca t ions of CU ind ices common in the l i t e r a t u r e . However, the economic ana lys i s a lso h igh l igh ts the competing and at times o f f s e t t i n g impacts on the leve l and d i r e c t i on of CU from forces such as non-stat ic expectat ions and d i f f e r e n t market s t ruc tu res . The information provided from the measure i s thus hard to untangle, and suggests d i f f i c u l t i e s involved in i n te rp re ta t i on of -175-TABLE III.1 — Cyc l i c a l Measures, T rad i t i ona l and Economic FRB Wharton Product i v i t y Tob in ' s q Capac CU CU Growth Rate Holland U t i l i ; Year (Manuf.) (Manuf.) (Business) Y/Y* 1 1954 80.3 88.1 1.4 0.69 not e; 1955 87.1 90.5 3.9 0.98 1956 86.4 87.9 0.3 0.97 1957 83.7 84.0 1.7 0.92 1958 75.2 74.2 2.4 0.83 1.106 1959 81.9 78.9 1.6 1.19 1.110 1960 80.2 76.9 2.5 1.15 1.171 1961 77.4 73.7 2.9 1.33 1.177 1962 81.6 76.5 3.6 1.31 1.197 1963 83.5 77.7 3.2 1.48 1.224 1964 85.6 79.5 3.9 1.73 1.226 1965 89.6 84.2 3.1 1.98 1.232 1966 91.1 88.2 2.5 1.66 1.214 1967 86.9 86.9 1.9 1.57 1.184 1968 87.1 89.2 3.3 1.68 1.178 1969 86.2 90.1 -0.3 1.50 1.169 1970 79.3 84.0 0.3 1.01 1.111 1971 78.4 82.6 3.3 1.21 1.110 1972 83.5 87.7 3.7 1.29 1.204 1973 87.6 92.9 2.5 1.10 1.240 1974 83.8 90.2 -2.4 0.54 1.092 1975 72.9 79.4 2.1 0.65 1.160 1976 79.5 85.5 3.2 0.68 1.259 1977 81.9 88.1 2.0 0.68 1.267 (Berndt) References: ( i ) Federal Reserve Board capac i ty u t i l i z a t i o n measure fo r manufacturing, Economic Report of the President (1981), Table B-43. ( i i ) Wharton capac i ty u t i l i z a t i o n measure f o r manufacturing, Economic Report of the Pres ident (1981), Table B-43. ( i i i ) Growth in labor product i v i t y for non-farm business sec tor , Economic Report of the President (1981), Table B-39. ( iv ) Tob in ' s q, D. Holland and Stewart Myers, American Economic Review, Papers and Proceedings, May 1980. (v) Economic measure of capac i ty u t i l i z a t i o n , Berndt (1980). -176-the concept of CU. The main goal of the essay i s thus to c l a r i f y these d i f f e r e n t dimensions, to provide a bas is for assessing the importance of var ious assumptions concerning economic s t ructure on movements in CU, to develop a conceptual framework to use for i n te rp re ta t i on of va r i a t i ons in a l t e rna t i ve c y c l i c a l i n d i c a t o r s , and to re l a te primal and dual measures of CU. In the next sect ion I b r i e f l y sketch the fundamentals of the "bas ic model" d iscussed in Essay 1, and ou t l i ne i t s usefulness in formulat ing and 4 analyz ing c y c l i c a l measures. I then consider how r e s t r i c t i v e foundations under ly ing t h i s basic model s p e c i f i c a t i o n have impl i ca t ions for measurement of c y c l i c a l v a r i ab l e s , and p a r t i c u l a r l y CU. S p e c i f i c a l l y , corresponding to the genera l i za t ions of Essays 1 and 2, va r i a t ions in der i va t ion and in te rp re ta t i on of CU measures are ana lyzed, g raph ica l l y and more formal l y , thereby expanding capac i ty u t i l i z a t i o n measures to r e f l e c t monopoly behavior and non-stat ic expectat ions , r e spec t i ve l y . The graphical cha rac te r i za t ion (Sect ion III) • provides motivat ion and in te rp re ta t i on fo r the corresponding ana l y t i c a l der i va t ion of the model (Sect ion IV). In the next sect ion (Sect ion V) I d iscuss empir ica l est imat ion of capac i ty u t i l i z a t i o n measures, both in terms of the quant i ty and the dual cos t s i de . F i n a l l y , in Sect ion VI , I present concluding remarks. -177-II. Economic Measures of CU and Other Cyc l i c a l Ind ica tors : An Overview CU i s fundamentally re la ted to var ious well-known economic i n d i c a t o r s . For example, the impl i ca t ions of va r i a t ions in capac i ty u t i l i z a t i o n fo r c y c l i c a l p roduc t i v i t y are often analyzed in terms of "non-optimal" l eve l s of inputs (or capac i ty ) i n the short run, where l eve l s of quas i- f ixed inputs or stocks d i f f e r from long run equ i l ib r ium l e v e l s . If one def ines capac i ty output Y* as the leve l of output which minimizes short run average to ta l cos ts (SRAC), then mu l t i f ac to r p roduc t i v i t y measures and short and long run p roduc t i v i t y growth rates fo r ind iv idua l inputs may be c y c l i c a l because of " d i s equ i l i b r i um" of actual output Y d i f f e r i n g from the capac i ty output Y*. For example, i f a shock such as an energy pr i ce increase changes economic capac i ty by s h i f t i n g the SRAC curve and thus a l t e r s the short run capaci ty u t i l i z a t i o n r a t i o , p roduc t i v i t y measures w i l l be a f f e c t ed . Mu l t i f a c to r p roduc t i v i t y growth w i l l be smal ler in the short than in the long run, because 5 with cap i t a l f i xed the f i rm i s constra ined in the short run by i t s f i xed inputs , whereas in the long run i t can adjust a l l inputs to long run equ i l i b r ium l e v e l s . It i s evident from th i s d iscuss ion that trends in capaci ty u t i l i z a t i o n can be formulated in terms of va r i a t ions in u t i l i z a t i o n of the f i r m ' s quas i-f ixed inputs . This a lso impl ies a c lose r e l a t i onsh ip with investment ana lys is and Tob in ' s q, noted by Yoshikawa (1980) and others : "The q theory, a l lowing the divergence between the value of c ap i t a l evaluated in the f i nanc i a l market and the pr i ce of cap i t a l goods, i s a theory which expla ins how investment (change in the cap i t a l stock) i s motivated by t h i s apparent short run d i s e q u i l i b r i u m . " C y c l i c a l i nd i ca to rs may thus a lso be charac ter ized by changes in the cos t of the cons t ra in t or shadow pr ice of quas i- f ixed inputs ; fo r c ap i t a l th i s shadow pr i ce i s the foundation of the concept of Tob in ' s q . 6 The value of q -178-i s usual ly def ined as the market value of a f i rm div ided by the replacement cos t of i t s physical c ap i t a l stock, q can be redef ined in terms of the shadow pr ice of i n s t a l l e d cap i t a l goods d iv ided by the tax adjusted pr i ce of un ins ta l l ed cap i t a l goods It can therefore be in terpreted as a " va lua t ion parameter" for the f i r m ' s cap i t a l s tock; net investment i s p o s i t i v e , zero , or negative when marginal q i s greater than, equal t o , or l ess than one. F i x i t y of stocks impl ies that marginal q i s not equal to one except in long run equ i l i b r i um, suggesting that the va r i a t i on of q from one depends on costs of adjustment. In such a framework, i t i s c l e a r that the value of q i s a funct ion of a l l va r i ab les a f f e c t i ng the f i rm , since var iab les such as energy pr i ce changes a f f e c t q through the i r impact on the shadow pr ice of c a p i t a l , q i s a lso c l ose l y re la ted to capac i ty u t i l i z a t i o n and labor product i v i t y measures through i t s impl i ca t ions fo r short run vs . long run subs t i tu t ion among inputs . S t a t i c models, although important, are inadequate to expla in these phenomena, because o f the lack of recogni t ion of s tocks . One can def ine a s t a t i c model as one having a l l inputs var iab le and a l l input flows adjust ing to t he i r long run equ i l ib r ium leve l s within a s ing le time per iod . Then, s ince a l l inputs are v a r i ab l e , rates of capac i ty u t i l i z a t i o n should not d i f f e r from un i t y , where capac i ty output (Y*) i s def ined as above, and the capac i ty u t i l i z a t i o n r a t i o i s Y/Y*. In terms of labor p roduc t i v i t y , i f a l l inputs were va r i ab l e , then with constant returns to scale any sudden increase in demand for output would be matched by a proport ional increase in the demand for a l l f ac tors of product ion, so that average labor product i v i t y would be unaf fected, and p rocyc l i c a l labor p roduc t i v i t y would be ru led out . F i n a l l y , i t i s not c l e a r why q would ever d i f f e r from one in a f u l l s t a t i c model. S p e c i f i c a l l y , when one in te rpre ts q as the r a t i o of the shadow pr ice to rental pr i ce of c a p i t a l , in s t a t i c models th i s r a t i o would always equal one. Thus, i t i s -179-c l e a r that s t a t i c models of economic opt imizat ion by the f i rm are unable s a t i s f a c t o r i l y to expla in c y c l i c a l phenomena such as both CU and q d i f f e r i n g from un i t y . Many attempts have been made to formulate dynamic models to take f i x i t y of stocks into account and thus to provide a theore t i ca l basis fo r c y c l i c a l phenomena, but these attempts have la rge ly been ad hoc. For example, the ex i s t i ng l i t e r a t u r e on short run labor p roduc t i v i t y has often postulated " p a r t i a l adjustment" o f actual to des i red stock l eve l s of inventory , l abor , or c a p i t a l , but seldom has t h i s empir ica l l i t e r a t u r e generated equations whose endogenous time paths were outcomes of an e x p l i c i t dynamic opt imizat ion process . Since there i s no economic st ructure to the adjustment process, most o f these ad hoc types of pa r t i a l adjustment models are not very s a t i s f a c t o r y . In con t ras t , the " s ta te of the a r t " l i t e r a t u r e which I have analyzed and extended in the previous Essays of t h i s d i s se r t a t i on can and to some extent has been used e x p l i c i t l y to consider c y c l i c a l phenomena with in a dynamic context . Th is dynamic l i t e r a t u r e i s useful for analyses of c y c l i c a l measures because of i t s e x p l i c i t incorporat ion of stocks and the impl ied adjustment process. As discussed at length in the preceding two essays, however, the ex i s t i ng models incorporate many assumptions that are too r e s t r i c t i v e a^ p r i o r i , some of which I have genera l ized . The extensions of those essays turn out to be extremely important components fo r ana lys is of the behavior of the f i rm in response to f l uc tua t ing exogenous va r i ab les . In order to c l a r i f y t h i s I b r i e f l y re-emphasize here the important c h a r a c t e r i s t i c s of the s t ructure of dynamic models and out l ine what extensions and refinements to th i s framework would be useful in order to analyze c y c l i c a l behavior. I then cons ider how the extensions a l t e r the in te rp re ta t ion and determinants of such measures as the shadow pr i ces of f ixed inputs (and -180-therefore q ) , optimal capac i ty and changes in labor p roduc t i v i t y , and f i n a l l y I proceed to analyze in more de ta i l the impl i ca t ions for CU. The previous two essays have been based on a model der ived from a dynamic opt imizat ion process by the f i rm . Th is framework i s based on costs of adjustment fo r quas i- f ixed inputs that induces slow adjustment by f irms to "opt imal " or "des i r ed " l e ve l s of the quas i-f ixed inputs . The model i s s t ructured so that one can e x p l i c i t l y der ive a system of short run demand equations for var iab le inputs ( u t i l i z a t i o n equations) and accumulation equation(s) for quas i- f ixed input(s ) based on an endogenous " f l e x i b l e acce le ra tor " or pa r t i a l adjustment process. Thus one can charac te r ize what the f i r m ' s optimal react ions to a temporary " d i s equ i l i b r i um" s i t ua t i on would be in the short run (with quas i- f ixed inputs f i x e d ) , long run (once quas i- f ixed inputs have reached t he i r long run l e v e l ) , and along the optimal adjustment path between these s ta tes . Short run, intermediate run, and long run cos t curves can be def ined analogously . Given th i s e x p l i c i t cha rac te r i za t ion of the f i rm ' s opt imizat ion process , one can der ive the f i r m ' s short run average to ta l cos t (SRAC) curve def ined by the given cap i t a l stock, the SRAC curve def ined by the "opt imal " c ap i t a l s tock, K*, the long run average to ta l cos t (LRAC) envelope and then ind ica te how the f irm w i l l move from one to the other . One can use knowledge of these cos t curves to charac ter ize the adjustment behavior o f the f i r m , o r , for example, to determine why the f i rm ' s capac i ty output Y* d i f f e r s from observed output Y. Hence one can e x p l i c i t l y determine an endogenous measure of capac i ty u t i l i z a t i o n , CU=Y/Y*. S i m i l a r l y , wi th in th i s same framework one can e x p l i c i t l y charac ter ize other c y c l i c a l measures, such as p roduc t i v i t y and Tob in ' s q. Measures of labor and mu l t i f a c to r p roduc t i v i t y w i l l vary between short run and long run -181-depending on the d i sequ i l i b r ium charac ter ized by the dev ia t ion o f capac i ty u t i l i z a t i o n from un i ty , and thus the r e su l t i ng dev iat ion of short run from long run input s u b s t i t u t i o n . As noted e a r l i e r , one can a lso def ine Tob in ' s q as the shadow value of cap i t a l d iv ided by the pr i ce of c a p i t a l ; thus q w i l l be unity only when CU i s un i t y , fo r given the cap i t a l stock the cap i t a l cons t r a in t w i l l be cos t l y only when the optimal output level cannot be at ta ined at minimum c o s t . 7 Thus in the dynamic opt imizat ion model a l l these measures are i n t e r r e l a t ed and are j o i n t l y determined. To discuss these i n t e r - r e l a t i onsh ip s , l e t us begin by assuming, as in Berndt, Morr ison, and Watkins (1981) (hereaf ter , BMW) that the f i rm can be represented within a dual cos t funct ion framework, character ized by long run constant returns to s ca l e . This i s the o r i g i na l " s ta te of the a r t " or basic dynamic model with constant returns to scale I shal l l a t e r genera l i ze . Such a formulat ion can be used to ca l cu l a t e an economic measure of optimal capac i ty output Y* as that Y which minimizes short run average tota l cos t . This al lows one to determine how Y* would be a f fec ted by changes in input pr i ces or other changes. S p e c i f i c a l l y , with cap i ta l as the only quas i-f ixed input , to determine Y* e x p l i c i t l y one d i f f e r e n t i a t e s average to ta l costs (equal to average var iab le costs plus average f ixed costs) with respect to Y and solves for Y* as the minimum point on t h i s curve, r e su l t i ng i n : 3.2.1) Y* = Y * ( K , K , P j , u k , t ) . I f one assumes a s p e c i f i c funct ional form for the var iab le cost funct ion G one can e x p l i c i t l y der ive the expression fo r Y*, thereby i nd i ca t i ng very c l e a r l y the fac tors determining short run capac i ty output. If G i s constant-returns-to-scale quadrat i c , say, as in Morrison-Berndt (1981) and cap i t a l i s the only quas i- f ixed input , with three var iab le inputs , labor (L ) , - 1 8 2 -energy (E ) , and mater ia ls (M), G can be represented a s : 3 . 2 . 2 ) G = L + P E E + P M M = Y ( a 0 + « 0 t t ^ E + < W - - 5 ( Y E E P 2 + Y M M P 2 ) + Y E I / E P M + a E t P E * + a M t P M * + a K ^ + * ^ Y K K ^ ^ + Y K K ^ ^ 2/Y) ^ + Y EK P E^ + Y MK P M^ + a Kt^* * The e x p l i c i t so lu t ion fo r Y* can be shown to be: 3 . 2 . 3 ) Y* = - ( Y K K K 2 + Y ^ K 2 ) / ( a K K + a K t K t + y^?^ K + y^f^ K + U^K) , which r esu l t s i n the capac i ty u t i l i z a t i o n measure: 3 . 2 . 4 ) CU = - Y ( c t K K + a K T K t + Y E | < P E K + Y M K P M K + U K K ) / ( Y K | < K 2 + Y ^ K 2 ) . This CU measure i s der ived and estimated in BMW for t he i r basic dynamic model, and has a lso been estimated by Berndt ( 1 9 8 0 ) . Using ( 3 . 2 . 3 ) , one can determine that an increase in the pr ice of a va r i ab le input such as energy w i l l increase Y* i f the var iab le input and cap i t a l are long run complements and decrease Y* i f they are subs t i t u t e s . I n t u i t i v e l y , i f cap i t a l and energy are complementary inputs in the long run, when the pr i ce of energy inc reases , the optimal cap i t a l stock decreases, given the same output. Thus, the ex i s t i ng cap i t a l stock i s cons is tent with a higher Y*. For example, i f energy and cap i t a l are complements, recent energy pr i ce increases could have increased Y*, which would reduce economic capac i ty u t i l i z a t i o n r a t i o s . T rad i t i ona l CU measures may not r e f l e c t t h i s type of phenomenon for the i r theore t i ca l economic underpinnings are unspec i f ied and 3 instead such measures tend to be ca l cu l a ted mechanical ly . For comparison with t r ad i t i ona l measures for the manufacturing sec tor , in Table 3 . 1 I reproduce the economic capac i ty u t i l i z a t i o n measures estimated by Berndt. Th is measure tends to fo l low patterns of the business cyc le f a i r l y w e l l , as charac te r ized by the other mechanical measures. However, the Berndt -183-capac i ty u t i l i z a t i o n measure i s always greater than one, implying a shortage o f capac i ty in the sense that the manufacturing sector appears always to be producing to the r i gh t of the minimum point on the SRAC curve. Th is in turn impl ies that in order to reduce un i t c o s t s , i t i s optimal to increase the cap i t a l s tock. Note that t h i s apparent chronic cap i t a l shortage could poss ib ly be due to a m is-spec i f i c a t i on of the model with e i the r the per fec t competit ion or s t a t i c expectat ions assumption. S p e c i f i c a l l y , in the f i r s t case a monopolist may not des i re as large a cap i ta l stock as a per fec t compet i tor , s ince the monopolist w i l l always be producing at a smal ler output leve l than that charac ter ized by the minimum po in t . A l t e r n a t i v e l y , with no forward-looking expectat ions the myopic f i rm i s always behind in ad just ing i t s cap i t a l stock, r e su l t i ng in observed "catch up" investment. I e laborate on these points l a t e r in t h i s essay. Using s im i l a r der i va t ions for adjustments in labor p roduc t i v i t y or q c h a r a c t e r i s t i c s of Tob in ' s q (q k ) w i th in t h i s e x p l i c i t dynamic opt imiza t ion framework i t can be shown that short run and long run p roduc t i v i t y growth r a t es , CU, the shadow pr i ce of cap i t a l and thus q k depend on and are a f fec ted by exogenous changes i n , say, Y, K, or P.. Thus, in the dynamic model one can determine fac tors a f f e c t i ng the divergence o f CU or q^ from un i t y , for the f i x i t y and endogenous adjustment of inputs provides an explanat ion as to why the f i rm may not be in long run equ i l i b r i um, and thus why in the short run Y* deviates from Y. The bas ic dynamic model a lso e x p l i c i t l y s p e c i f i e s how the f i rm w i l l adjust toward the long run equ i l i b r i um, given an exogenous shock. Economic analyses of these in te r- re l a ted c y c l i c a l i nd i c a to r s , t he i r in te rpre ta t ions and explanat ions , are therefore obviously f a c i l i t a t e d by use of the dynamic model. However, as emphasized throughout th i s t h e s i s , the resu l t s and in te rp re ta t ions depend c r i t i c a l l y on the given assumptions. There are many -184-theore t i ca l assumptions inherent in t h i s basic dynamic model which are too r e s t r i c t i v e a p r i o r i . For example, the basic dynamic model cannot show how p roduc t i v i t y , q k , or Y* depend on expectat ions , s h i f t s in the demand curve and/or corresponding movements along i t . Such phenomena are important determinants of c y c l i c a l measures. A less r e s t r i c t e d s p e c i f i c a t i o n of the f i rm ' s opt imizat ion dec i s ions , such as those I der ived in Essays 1 and 2, may therefore be very useful fo r a more de ta i l ed and r i che r ana lys is of c y c l i c a l i nd i c a to r s . To extend the bas is of c y c l i c a l measurement wi th in an economic opt imizat ion model, I w i l l therefore proceed to analyze the measurement and in te rp re ta t i on of CU ind i ca to r s corresponding with these genera l i za t ions . The simultaneous incorporat ion of genera l i za t ions of the assumptions of s t a t i c expectat ions , per fec t compet i t ion, cost min imizat ion, and costs of adjustment on net rather than gross investment resu l t s in an extremely complex model s p e c i f i c a t i o n . The sequence of reasoning in th i s Essay w i l l therefore fo l low the same general d i v i s i ons as the previous two essays. S p e c i f i c a l l y , extensions of the cha rac te r i za t i on of CU to monopol ist ic behavior and to non-stat ic expectat ions w i l l be pursued i n d i v i d u a l l y , f i r s t g raph i ca l l y to emphasize the adaptations in model der i va t ion and in te rp re ta t i on fo r the gene ra l i za t i ons , and then more forma l l y . Although my development of capac i ty u t i l i z a t i o n impl i ca t ions suggests analogous charac te r iza t ions fo r q and p roduc t i v i t y , fu r ther in-depth ana lys i s of these other app l i ca t ions w i l l be l e f t fo r future research. -185-III. A Graphical Der ivat ion and Interpretat ion of Economic Capacity U t i l i z a t i o n Measures: Extensions to the Monopoly and Non-Static Expectat ions Models Theoret i ca l and empir ica l imp l i ca t ions fo r measurement o f c y c l i c a l i nd i ca to r s in the basic dynamic model vary according to the s t ructure imposed. To c l a r i f y t h i s I f i r s t d iscuss the exogenous output cost minimizat ion case and then examine resu l t s of separately genera l i z ing var ious assumptions. The concept of a capac i ty u t i l i z a t i o n (CU) measure in the cost funct ion case i s qui te s t ra ight forward . CU i s b a s i c a l l y a short run not ion , cond i t iona l on the f i rm ' s quas i-f ixed input s tocks. In the r e s t r i c t e d cost funct ion case , the f i rm minimizes var iab le c o s t s , cond i t iona l on Y, P., and x. Given long run constant returns to s ca l e , capac i ty output Y* can be def ined as that level of output for which average tota l cost i s minimized, as in (3.2.1) above. As i s i l l u s t r a t e d in F igure 11I- l , th i s def ines the tangency between the long run average tota l cost (LRAC) and short run average to ta l cost (SRAC) curves . In the more general case of non-constant re turns , Y* can s i m i l a r l y be def ined as the tangency between the long run and short run average to ta l cos t curves ; c l e a r l y in such a case Y* does not necessar i l y correspond to the minimum point on the short run average tota l cos t (SRAC) curve. Y* thus depends on va r i a t ions in the exogenous va r i ab l es . G raph i ca l l y , t h i s dependence can be expressed, for example, as whether a change in an exogenous va r i ab le s h i f t s the minimum point of the SRAC curve to the r i gh t ( increas ing Y*) , to the l e f t (decreasing Y*) , or j u s t s h i f t s the average cos t curve upward without a f f e c t i ng Y*. -186-Figure III-l Figure 111-2 SRMC -187-Since these cos t curves r e f l e c t to ta l c o s t , t he i r pos i t i on i s determined by the f i xed cap i t a l s tock; expansion of the cap i t a l stock s h i f t s the curve to the r i g h t . Note a lso that the long run equ i l ib r ium point fo r the f i rm occurs when cap i t a l has been adjusted so that the tangency po int of the SRAC and LRAC curves co inc ide with the given output l e v e l . At th i s po int the CU measure equals un i t y , as in the s t a t i c case . However, i t can be argued that t h i s cha rac te r i za t i on of the f i rm i s too r e s t r i c t i v e in the sense that a f i xed output leve l const ra ins the behavioral responses of the f i rm more than i s des i rab le a p r i o r i . 111.A. P r o f i t Maximization and CU Measurement The incorporat ion of an endogenous output level along with the behavioral assumption of p r o f i t maximization creates important changes in the i n t e rp re ta t i on and der i va t ion of the capac i ty output measure Y * . Assume CU i s s t i l l def ined as some actual leve l of output r e l a t i v e to an optimal leve l of output Y * . Now, however, optimal output w i l l be endogenous. Say Y Q * can be de f ined , analogously to Y * above, as the tangency between the SRAC and LRAC c u r v e s . 1 G Although i t i s not immediately c l e a r why the f i rm would not simply move to reach the "optimum" Y at a l l times s ince Y i s a choice var iab le fo r the f i r m , a useful optimimum output measure can be der ived for such a case. Assume that o r i g i n a l l y the competi t ive f i rm i s at a long run equ i l ib r ium point where the actual c ap i t a l stock i s cons i s ten t with p r o f i t maximization given the expected values of the exogenous va r i ab l e s , and thus co inc ides with the "des i r ed " or steady state stock l e v e l . Then an exogenous shock occurs which changes the industry demand and thus the p r o f i t maximization cond i t ions fo r the f i rm by changing the exogenous e f f e c t i v e demand pr ice from PQ to P,. The pos i t i on of the SRAC curve then corresponds to the previous -188-ca l cu l a t i ons of optimal K*, which no longer are v a l i d . The new short run p r o f i t maximization po int given the o ld K* i s Y^* (see F igure 111-2), which represents a s i t ua t i on where there are excess p r o f i t s with pr i ce equal to short run marginal costs but with average cost less than average revenue (=P 1 ) . However, given the e x i s t i n g cap i t a l s tock, K, the output leve l > corresponding to an equ i l ib r ium would be the minimum point of the SRAC curve. This i s the sense in which the prev ious ly def ined Y Q * was an "optimum" output l e v e l . Th is d iscuss ion h igh l igh ts the i n te rp re ta t i on of the "capac i ty " output leve l in terms of a steady state or long run where there are no incent ives to move. I t i s not j u s t i f i a b l e to extend t h i s ana lys is to the indust ry , fo r the long run industry capac i ty w i l l of course adjust given f u l l mob i l i t y of f i rms . However, in the short-run sense industry capac i ty i s f i x e d , for there i s no e x i t or entry of f i rms , and no change in cap i t a l stock for e x i s t i n g f i rms. Thus Y Q * can be def ined as the short run "capac i ty " output and Y-^ * as the short run "opt imal " output. These two concepts w i l l d i f f e r with "temporary" equ i l i b r ium from f ixed fac to r l e v e l s . Given these d e f i n i t i o n s , one in te rpre tab le CU measure, based on the actual maximum p r o f i t pos i t i on Yj^*, can be ca l cu l a ted as Y]*/Y Q *. An a l t e rna t i ve measure, more cons i s t en t with the previous cost funct ion case i s Y / Y Q * , where Y i s actual output. Th is measure would be the same as the former in the absence of e r ro r s in p r o f i t maximizat ion, i . e . , the divergence of the two must be explained by e r ro rs of the f i rm in p r o f i t maximization. The p r o f i t maximizing cha rac te r i za t ion o f the f i rm ' s dec is ions i s s t i l l in a sense too r e s t r i c t i v e a s p e c i f i c a t i o n , p a r t i c u l a r l y for the ana lys i s of the i ndus t r i a l l e v e l , as I d iscussed in Essay I and in Morrison (1981). S p e c i f i c a l l y , in t h i s case the competi t ive f i rm i s not react ing according to a -189-demand curve but simply to a given pr i ce that i s determined by demand at the industry l e v e l . Fu l l industry adjustment i s not r e f l e c t ed in the f i rm ' s behavioral pa t te rn . Th is formulat ion remains, however, cons i s ten t with the r e s u l t , based on the simpler s t a t i c expectat ions cost minimizat ion cha rac te r i za t ion that capac i ty output lags behind the actual output l e v e l , as permanent shocks in demand become observed. Further i n t e r e s t i ng va r i a t ions in der i va t ion and in te rp re ta t i on o f CU measures a r i se in the monopol ist ic case, charac ter ized by the model in Essay 1. Here one can assume the behavior of the monopolist i s equiva lent to the behavior of the en t i re indust ry , or at l e as t that the "monopolist" i s a large enough part of the industry to r e f l e c t industry behavior cons i s t en t l y . In t h i s case one can s t i l l def ine and compare a Y Q * and a Y^*. However, the s t ructure of the ana lys i s i s d i f f e r e n t . Y Q * i s s t i l l in a sense a long run notion of capac i t y , where long run marginal revenue ( L R M R ) i s equal to long run marginal cost (LRMC). As seen in F igure III-3, Y Q * i s the output the f i rm would choose i f the demand curve were stable and were expected to remain stable where L R M R = L R M R Q . However, given an exogenous s h i f t in the demand 12 curve (see the new marginal revenue curve L R M R Q ) and f i x i t y of c ap i t a l the f i rm w i l l want to produce a new output Y j * . Thus two CU measures can be de f ined , Y/Y Q* and Y - ^ / Y Q * , analogously to the per fec t compet i t ion p r o f i t maximization case . The f i rm would be at a long run "optimum" production l e v e l , i . e . , i t would not have any incent ive to move from th i s point i f the current K = K * . Thus one can def ine Y Q * as the steady state Y cons is tent with th i s K * . Th is d e f i n i t i o n i s analogous to the cos t minimizat ion d e f i n i t i o n where the minimum point on the SRAC curve would charac ter ize the long run optimum leve l of output i f the actual cap i t a l stock K were equal to K * . However, in the monopoly case the Y * po int w i l l be to the l e f t hand side of the minimum -191-point of the S R A C curve which represents a point of short and long run increas ing returns to sca le . It should be noted, however, that when K^K* and therefore Y^Y* (the r e s u l t of an unant ic ipated exogenous shock in a previous time per iod which has not ye t been f u l l y accommodated) the current short run p r o f i t maximizing Y=Y^* could end up on the r i gh t hand s ide of the minimum po in t of the S R A C curve. Th is Y^* w i l l in turn be d i f f e r en t from actual Y because of s tochast i c er rors in p r o f i t maximization. Two comparable measures 13 are therefore imp l i ed , Y/Yj* and Y Q */Y j * , where va r i a t ions in the measures can be determined as e x p l i c i t funct ions of the f i r m ' s dec i s ion v a r i ab l e s , so that the e f f e c t s of exogenous changes on capac i ty output and thus C U can be der i ved . The i n t e rp re t i v e ana l ys i s can be expanded to c l a r i f y in what sense Y Q * corresponds to a long run equ i l ib r ium po in t . In F igure 111-4, assume that the monopolist i s a t long run equ i l ib r ium with a l l exogenous var iab les at steady state va lues . Given the long run curves L R M C and L R M R Q , the monopolist w i l l produce at the po int where these i n t e r s e c t . I f the demand curve s h i f t s out exogenously and unexpectedly, the new long run equ i l ib r ium point for the monopol ist , given that he has s t a t i c expectat ions , w i l l be at the i n t e r sec t i on of L R M R J and L R M C , or at an output leve l of Y 2 * as shown in F igure 111-4. However, in the short run, the f i rm ' s myopic behavior has resu l ted in a lower cap i t a l stock than i s required to take immediate advantage of the increased potent ia l stream of p r o f i t s cu r ren t l y ava i l ab le from the increased demand. The given cap i t a l stock def ines the curves S R M C Q and S R A C Q , so production in the short run w i l l be at the L R M R ^ S R M C Q po in t , or at output leve l Y l * * Y 0 * ' n o w e v e r » remains the "optimal capac i ty " output in the sense that i f the current K represented an optimum, as i t would i f the f i rm had been co r rec t about the long run values of i t s dec is ion va r i ab l e s , there would be no incent ive to vary from th i s output l e v e l . -192-Capacity u t i l i z a t i o n i s therefore viewed as a short run phenomenon, with 14 cur rent exogenous var iab les given , such that temporary equ i l ib r ium rather than f u l l equ i l ib r ium ex i s t s and Yj* and Y Q * do not co i n c i de . The p r o f i t maximizing monopolist therefore has incent ives to move, fo r the f i xed f a c to r cons t ra in t i s cos t l y in terms of p r o f i t a b i l i t y . Th is i s cons i s ten t with the Le Cha te l i e r concept that a constra ined so lu t ion in the short run w i l l not be pre fer red to the long run unconstrained so l u t i on . Thus the f i rm w i l l expand cap i t a l to reach a short run average cos t curve , SRACp which corresponds to Y 2 * . As the cap i t a l stock i s gradual ly sh i f t ed toward th i s new equ i l i b r ium l e v e l , output p r i ce and va r i ab le input demand adjust to compensate fo r the gradual ly c l o s ing gap between capac i ty and demand. An a l t e rna t i v e motivat ion fo r understanding the concepts involved in measuring CU corresponds more d i r e c t l y to the mathematical der i va t ion of the next sec t ion . At the i n i t i a l equ i l ib r ium po in t , the f i r m ' s shadow va luat ion of c ap i t a l w i l l not deviate from the actual p r i ce of c a p i t a l . However, when demand increases , the f u l l shadow value of cap i ta l w i l l exceed the market p r i ce of c a p i t a l , due to the increased potent ia l p r o f i t a b i l i t y of the incremental cap i t a l stock and the f i xed cap i t a l cons t ra in t which r e s t r i c t s the f i rm from immediately adjust ing the cap i t a l stock to the point where th i s potent ia l i s captured (LRMR^LRMC). Thus there w i l l be an incent ive for the f i rm to increase investment in order to decrease costs in the long run, which impl ies a s h i f t of SRAC to the r i gh t over t ime. This w i l l continue un t i l the shadow value of cap i t a l evaluated at the then current value of K i s equated to the market pr ice of c a p i t a l , which w i l l be true at the long run equ i l ib r ium charac ter ized by LRMR^LRMC. 1 5 Th is scenar io can be used to h igh l i gh t the l inkages between the CU concept and the shadow pr ice of c ap i t a l or q k . The long run marginal cost curve takes a l l costs into account inc lud ing costs of adjustment of cap i t a l and -193-therefore ind ica tes what the long run lowest cos t cap i t a l accumulation path i s f o r the f i rm at d i f f e r e n t long run marginal revenue l eve l s and the given cap i t a l stock. In the long run the monopolist always wants LRMC=LRMR because at t h i s point long run p r o f i t s are maximized. At the chosen output leve l K*=K, and at the given cap i t a l stock leve l Y=Y* so there i s no incent ive to move from these va lues . In t h i s sense the ex i s t i ng Y and K are cons i s ten t with each other and therefore with a steady state so CU=q k=l. Thus both the short run and long run behavior of the monopolist can be in te rpre ted in terms of CU or wi th in the dynamic framework with one quas i-f ixed input . The various model formulat ions analyzed above, p a r t i c u l a r l y in the monopol ist ic case , al low the behavioral assumptions of the f i rm to be more general and thus more r e a l i s t i c than in the basic dynamic model of BFW. However, in some ways even these more general models are s t i l l too r e s t r i c t i v e , p a r t i c u l a r l y with respect to an t i c ipa ted changes in future exogenous va r i ab les . I I I.B. Non-Static Expectat ions and CU Measurement If the model were extended to ce r t a i n non-stat ic expectat ions , then in some sense the f i rm would always be producing at i t s "optimum" l e v e l . Any va r i a t i on from t h i s would be in te rpre ted as the r esu l t of disturbances from er rors in the opt imizat ion process of the f i r m . This does not, however, mean that the f i rm would always be producing at i t s s t a t i c or current valued optimal l e v e l . Even i f expectat ions were pe r f ec t , given costs of adjustment sudden changes in exogenous var iab les cannot be accommodated quick ly and c o s t l e s s l y as they occur . Thus, in the current time pe r iod , the f i rm w i l l not necessar i l y be at a s t a t i c optimal po int in terms of the tangency of the SRAC -194-and LRAC curves, un less , in some sense, the cos t curves are def ined in terms of present value l eve l s of output and costs i n c lus i ve of adjustment cos t s . This i s an important d i s t i n c t i o n . Therefore I w i l l b r i e f l y d igress to d iscuss fundamental concepts involved in the app l i ca t i on of non-stat ic but per fec t expectat ions (corresponding to Essay 2) to the CU conceptual framework. If the f i rm an t i c ipa tes non-constant future paths of exogenous var iab les such as p r i c e s , output, or demand, in order to determine the cur rent investment path the en t i r e future time path must be taken into cons ide ra t ion . As a simple example, assume that at time tg the p r o f i t maximizing monopolist perceives a permanent upward s h i f t in the demand curve at time t^, with a l l other exogenous var iab les remaining constant fo r a l l t ime. With no costs of adjustment, the optimal c ap i t a l stock would increase to from K Q* (corresponding to the o r i g i na l demand curve) instantaneously at time t^, implying an i n f i n i t e investment rate at that t ime. With adjustment c o s t s , however, given that at the current time per iod tg the f i rm expects the shock to occur at t p the investment rate fo r the f i rm w i l l increase immediately from the investment rate which i s optimal under constant s t a t i c expectat ions fo r a l l exogenous v a r i ab l e s , to a la rger current net investment rate fo r a l l time periods between tg and t^. Thus the f i rm w i l l approach the leve l gradua l l y , s t a r t i ng at time t g . Current investment, the re fore , i s more cos t l y than i s optimal in the current sense. However, these expenses reduce some future costs which would r e s u l t from adjustment that would have to be ca r r i ed out more r ap id l y , and thus at higher present value cos t i f there were no an t i c i pa t i on of the s h i f t a t t j . In terms of the flow of p r o f i t over t ime, th i s impl ies that the f i rm w i l l be able to benef i t from having an t i c ipa ted the exogenous change. Any an t i c ipa ted exogenous shock that increases the present value of the revenue -195-accru ing over time to the marginal un i t of cap i t a l without a f f e c t i n g i t s cos t makes investment more p r o f i t a b l e . F igure 111-5 i l l u s t r a t e s the importance of future expectat ions on current investment by cha rac te r i z ing d i f f e r e n t investment paths t j which depend on var ious expectat ions about the t iming of the demand increase , represented by the time supersc r ip t " i " . These investment paths can be charac te r ized in general terms. Although the optimal cap i t a l stock w i l l not be reached at time t 1 , the cu r ren t l y optimal investment path (based on time t j current va r iab les ) w i l l be reached at time t j , so that the remaining adjustment proceeds from t j onward with the marginal costs of adjustment balanced before and a f t e r the shock. In add i t i on , the c lose r the current time per iod i s to the time of the expected shock, the stronger i s the e f f e c t on current investment p lans. With s t a t i c expectat ions , in con t ras t , a shock at time t j w i l l not be accommodated un t i l time t j i s reached. With t h i s unexpected shock the f i rm i s not e f f e c t i v e in maximizing i t s flow of net revenues over time and therefore i s not on the optimal path in the current or present value sense once the shock takes p lace , as seen from time tg in the per fec t fo res ight case . However, as seen from time t j , the f i rm i s cu r ren t l y opt imiz ing over a l l future time periods as the past values represent sunk cos t s , and between time t Q and t^ the f i rm i s opt imiz ing given current va lues . From time t j onward, there fore , the f i rm ' s dec is ion c r i t e r i o n appears the same in both the s t a t i c expectat ions and non-stat ic expectat ions cases. During the per iod t Q to t | , in the non-stat ic expectat ions case the f i rm i s on the optimal path in terms of maximum present value over time but i s 16 not on the optimal path in terms of cur rent v a r i ab l e s . This i s r e f l e c t ed in the adjustment paths p ic tured in F igure 111-5 for d i f f e r e n t t iming of an increase in output p r i c e , for a l l r e su l t from actual patterns of exogenous Figure II1-5 i 1 1 1 1 1 1 1 ! 1 I 1 1 I ! i i \ i i i i i i i i i i -197-var iab les that are i den t i ca l between time t Q and t j . Therefore , a f t e r time t j , although the dec is ion c r i t e r i o n w i l l be the same in both s t a t i c and non-stat ic expectat ions scenarios given current exogenous va r i ab l e s , observed behavior w i l l d i f f e r as some adjustment in the "g iven" var iab les w i l l have already taken place and thus current cos ts w i l l be lower. These resu l t s have been der ived more formal ly in Essay 2 by e x p l i c i t l y taking in to account the time-dependence of the "des i r ed " cap i t a l t a rge t . Th is model w i l l be referenced f o r development of the ana l y t i c s and empir ica l r e su l t s in the next sec t ions . The model can be i n t u i t i v e l y motivated, however, in the context of the analogous d i sc re te model der i va t ion in N i cke l l (1978). His i n te rp re ta t i on o f the non-stat ic expectat ions case , as s tated i n Essay 2, i s that in a general sense the f i rm should be aiming at "some sor t of average of a l l the d i f f e r e n t future ' de s i r ed ' c ap i t a l s tocks " , so that " ins tead of the f i rm aiming at a simple ' d e s i r ed ' c ap i t a l stock, K*, i t aims a t the des i red cap i t a l stock fo r the next per iod plus an exponential weighted sum of the d i f fe rences between the des i red cap i t a l stock next per iod and the d i f f e r e n t des i red cap i t a l stocks fo r a l l fu ture p e r i o d s " 1 7 Th is model a l so f a c i l i t a t e s in te rp re ta t ion of the d i s t i n c t i o n made e a r l i e r between the optimal current versus present value dec is ions of the f i r m . S p e c i f i c a l l y , the form of the investment equation der ived i n Essay 2 i s : 3.3.1) I(t) = xK(t)- xK*(t)+ (1/G^)L e ( x " r ) ( s " t ) ( ( G K K + r G | ^ | < ) ( ( K * ( s ) - K * ( t ) ) + G^P(s) + G^ YY(s)) ds = |x| (K* ( t ) - K(t) + J) =UI(K**(t) - K(t ) ) where K*(s) i s the "des i r ed " cap i t a l stock def ined fo r p r i ces and demand l eve l s at time s and |xjis the pa r t i a l adjustment parameter. It i s c l ea r that i f a l l exogenous var iab les were constant , K*(s)=K*(t)=K*, so J=0 which reduces -198-to the standard f l e x i b l e acce le ra tor model. Thus, the current optimum value i s given by K*(t) , and the present value maximizing " ta rget " stock leve l i s given by K**(t)=K*(t)+J, which by d e f i n i t i o n puts the f i rm at an intertemporal optimum by taking into account the present value of a l l future exogenous changes fo r the f i rm . Capacity output d e f i n i t i o n s and in te rp re ta t ions vary with t h i s more complex opt ima l i t y representa t ion . With non-stat ic expectat ions the SRAC curve and thus optimal capac i ty w i l l depend on the adjusted cap i t a l stock, K**(t) , instead of K*(t) . To motivate the CU der i va t ion in t h i s case , consider a cos t minimizing f i rm at time t Q expecting with per fec t ce r ta in t y an increase in given output a t time t j . As above, assume that the f i rm w i l l increase investment a t a l l times previous to t j compared to the case of constant output expectat ions, so that the current cap i t a l stock i s above the cur rent l y optimal stock between tg and t p Thus, at any time between time LQ and t^, assuming that before time tg the f i rm was in long run equ i l ib r ium with exogenous Y equal to capac i ty output Y*, the current c ap i t a l stock w i l l def ine a SRAC curve which reaches a minimum to the r i gh t of the given output l e v e l . This represents a point of excess capac i ty in current terms, as the current-valued diagram w i l l ind ica te a divergence between Y* and Y. It i s , however, the optimum leve l of output given a present value opt imiz ing dec i s i on , which i s why the current measure of capac i ty u t i l i z a t i o n may diverge from unity in the short run even when the f i rm i s at a true intertemporal optimum. Neglect of expectat ions of future changes in exogenous va r i ab les impl ies that K**=K* in the f i rm ' s decision-making process. Then even i f the current diagram ind ica tes equa l i t y between Y* and Y, the SRAC curve and the resu l t i ng measure of Y* w i l l be to the l e f t of the intertemporal optimal point at any -199-time between t Q and t ^ Thus, i f output i s tending to increase and i s unant i c ipa ted , the f i rm w i l l always be lagging behind in terms of adjustment of capac i t y . The seemingly paradoxical r esu l t s reported in Berndt (1980) and Berndt Morrison and Watkins (1981) can be in terpre ted in th i s manner. The re levant CU measure in th i s case i s not immediately c l e a r . I f capac i ty output in F igure 111-6 i s de f ined , as before , as the minimum point on the SRAC curve and i s compared to current output, the r e su l t i ng current CU measure w i l l d i f f e r from unity between time tg and t p even though i t i s optimal in the intertemporal present value maximizing sense. I f output i s def ined as a present value notion of a future path of output l e v e l s , the optimal capac i ty measure and actual output w i l l always co inc ide and CU w i l l be un i t y . CU measures, however, must be based on current observat ion. Y** can therefore be in te rpre ted as a cha rac te r i za t ion of the present value of expected output, and the d i f fe rence between actual Y and Y** can be in te rpre ted as represent ing the impact on capac i ty output of forward-looking 18 expectat ions . Th is i s in d i r e c t cont ras t to the s t a t i c expectat ions case, where the d i f f e rence between Y and Y* i s a t t r ibu ted to previous shocks that were not expected and thus must be accommodated ex post . The d i f fe rence between these concepts i s i l l u s t r a t e d in F igure 111-7. Note that in general K may not co inc ide with K** and thus Y may not be equated to Y** by time t ^ although the optimal investment path using current va r i ab les i s reached at time t^. Th is i s a r e su l t of the p o s s i b i l i t y of optimal t rade-of fs of costs of adjustment before and a f t e r the shock, given per fec t f o r e s i gh t , so that the f i n a l equ i l ib r ium values may not be reached u n t i l , say, t^. Also note that t h i s i s a very s i m p l i f i e d case with one permanent shock in one exogenous va r i ab l e . In r e a l i t y the paths of exogenous var iab les w i l l a l l be changing over time and not a l l changes w i l l be permanent. Th is great ly complicates the ana l y s i s . -200-Figure 111-6 t c) Y l t 1 i 1 - 1 1 l actual Y | » l 1 1 1 1 1 -201-Further complex i t ies a r i s e when expectat ions are not pe r f e c t . Then the va r i a t i on between Y * * and Y cannot a l l be a t t r ibu ted to cor rec t forward looking expectat ions and therefore Y * * cannot be in te rpre ted as a present value of a stream of output. Instead, part of t h i s divergence must be accounted for by previous e r rors in adjustment, analogously to the constant expectat ions case when there i s an unant ic ipated shock. Thus, the actual CU r a t i o w i l l be bounded by the Y * * / Y and Y * / Y va lues , where the former w i l l diverge from unity according to per fec t forward-looking expectat ions and the l a t t e r value w i l l vary because of e r rors in expectat ions . Such competing in f luences cause i n t e rp re ta t i on of actual observed CU measures to be necessar i l y ambiguous. Since the optimum capaci ty in t h i s case i s l a rger than in the previous ana l y s i s , there would be less chance of observing apparent chronic shortages of c a p i t a l . The f i rm ' s behavior i s l ess l i k e l y to be charac ter ized by constant "catch-up" investment i f K * * rather than K * i s the target c ap i t a l stock leve l for the f i r m , as the f i rm ' s adjustment of i t s cap i t a l stocks w i l l be an t i c i pa to r y , not always re tarded. The r e su l t i ng estimated CU measure could therefore end up at e i the r s ide of un i t y . Although the model in Essay 2 i s based on cos t min imizat ion, i t i s informative b r i e f l y to consider the monopol is t ' s behavior in the non-stat ic expectat ions case . As with the cos t funct ion case, the monopolist w i l l look ahead and adjust the paths of his contro l var iab les to maximize the present value of p r o f i t s . Thus, given an exogenous shock at time t , the monopolist w i l l not remain at the leve l of c a p i t a l , K Q , de f in ing the SRAC curve corresponding to Y Q * , un t i l time t^. At time t g , when an upward s h i f t , say, in marginal revenue from L R M R Q to L R M R ^ i s expected to occur at time t^, the monopolist w i l l immediately begin to expand cap i t a l and thus capac i t y . S p e c i f i c a l l y , he w i l l invest to s h i f t the SRAC and SRMC curves -202-toward SRAC^ and SRMC^ before the demand s h i f t occurs , producing at each po in t between t Q and where S R M C = L R M R Q (see Figure 4 ) . As the cap i t a l stock gradual ly i s adjusted corresponding to the expected demand change, output p r i ce as well as var iab le input demand adjust to compensate for the gap between current observed capac i ty and demand. Some short run adjustment to compensate fo r the f i x i t y of cap i t a l can be made by a l t e r i n g the (pe r fec t l y var iab le ) labor input (as in the other models), but with the monopoly model fur ther " f i n e tun ing" can be done by moving along the demand curve. Over the adjustment pe r i od , as the SRAC curve s h i f t s to the r i g h t , the SRMC curve w i l l i n t e r sec t the o r i g i na l MR curve at a po int 19 corresponding to inc reas ing ly lower pr i ces and higher output l e v e l s . Thus output p r i ce drops un t i l the demand s h i f t and then spikes up, perhaps temporari ly even higher than i s optimal in the long run i f f i n a l adjustment to the steady state value i s "spread out" un t i l a f t e r the shock to t 2 » This i s in d i r e c t cont ras t to the s t a t i c expectat ions case where output p r i ce and va r i ab le input demand l eve l s are constant at long run equ i l ib r ium leve ls unt i l the exogenous (unantic ipated) s h i f t occurs . This s t a t i c expectat ions case would r e s u l t in a sudden output p r i ce increase and then a gradual decrease towards the equ i l ib r ium level as capac i ty expands to correspond to the new demand l e v e l . Short run va r i a t ions in the var iab le inputs w i l l of course a lso take place along the adjustment path r e f l e c t i n g subs t i tu t ion fo r the f i xed f a c to r . Thus, both s t a t i c and non-stat ic expectations cases produce v o l a t i l e behavior fo r the monopol ist ic f i r m . However, they pred ic t very d i f f e r e n t paths of behavior, in p a r t i c u l a r fo r determination of output pr i ce and labor (var iab le input) demand adjustment. The above ana lys i s can a l so be r e in te rp re ted , analogously to the s t a t i c expectat ions case , in terms of the shadow value of c a p i t a l . Once the future -203-demand change i s an t i c i pa t ed , the current shadow value of cap i t a l increases because the an t i c ipa ted p r o f i t stream from the marginal un i t of cap i t a l has increased . The to ta l shadow value i s equal to the cur rent s t a t i c marginal value given a l l exogenous var iab les plus the extra discounted revenue poss ib le from that extra un i t of cap i t a l stock provided by the incremental demand in the fu ture , analogous to the K** versus K* determination equation represented by (3.3.1) to (3.3.4) Thus, the shadow value of cap i t a l and q k are l a rger at time than they would be with s t a t i c expectat ions , which encourages extra an t i c ipa to ry investment toward the f i na l equ i l ib r ium l e v e l . S i m i l a r l y , i f reduct ions in future demand are an t i c i pa t ed , q^ based on non-stat ic expectat ions w i l l be smal ler than q^ based on s t a t i c expectat ions . The capac i ty output leve l can therefore be in terpre ted as the s tat ionary leve l of Y at each K as i t adjusts to i t s new equ i l ib r ium l e v e l ; i t i s the point where the current SRAC curve, def ined by the given K which i s at an inter tempora l ly but poss ib l y not cu r ren t l y optimal l e v e l , i s tangent to the LRAC curve. Since th i s cap i t a l leve l takes the overa l l present value into account, the corresponding capac i ty output r e f l e c t s the adjustment of cap i t a l toward K** rather than K* and may not correspond to the current "opt ima l " 20 output leve l except in the steady state where K=K** and thus Y=Y**. The d i scuss ion to t h i s point h igh l igh ts two important facets of the i n v e s t i g a t i o n . F i r s t , there i s a great d i f f e rence in pred ic ted behavior depending on whether expectat ions are s t a t i c or non-stat ic . These imp l i ca t ions of the d i f f e r e n t expectat ions models are i l l u s t r a t e d q u a l i t a t i v e l y in F igure II1-8. For purposes of comparison, p roduc t i v i t y i s indexed to unity in time t g , and returns to unity in the long run implying 21 constant returns to s ca l e . In t h i s simple case analyzed here, a l l adjustment in the s t a t i c expectat ions case comes " a f t e r the f a c t " — a f t e r the Figure II1-8 CU, q . , and produ " I r (a) General s t a t i c expectat ion resu l t s with an exogenous increase in demand at time t.j -205-exogenous stock has been observed—whereas in the non-stat ic expectat ions case 22 adjustment i s " forward-looking" . Thus the CU in te rpre ta t ions are very d i f f e r e n t in the two cases . In the s t a t i c expectat ions case Y* i s der ived given constant current va r i ab l es . Any divergence from th i s Y* value (and thus the magnitude and s ign of the CU measure) i s assumed to be the r e su l t of previous unant ic ipated shocks that have not ye t been f u l l y accommodated. In the non-stat ic formulat ion a s im i l a r long run Y* can be der ived fo r constant expectat ions , and a Y** can be ca l cu l a t ed which var ies from Y* as a r e su l t of c ap i t a l adjustment corresponding to forward-looking expectat ions . The dev ia t ion of the actual Y from these va lues , which defines the CU measure, w i l l therefore in general be a complex combination of these two shocks, previous disturbances and adjustment corresponding to forward-looking 23 expectat ions , a l l of which w i l l be d i f f i c u l t to untangle. F i n a l l y , qua l i t a t i v e changes in CU in response to an exogenous outward s h i f t in the demand curve at time t=t^ as viewed at time t Q can be summarized fo r the various cases discussed above. Given the i n t e r r e l a t i onsh ips among the c y c l i c a l i nd i c a to r s , one can use these r esu l t s to derive the corresponding impl i ca t ions on q and labor product i v i t y measures. S p e c i f i c a l l y , the above d iscuss ion impl ies that industry CU measures exh ib i t s im i l a r trends fo r a l l opt imizat ion behavior assumptions—cost min imizat ion , p r o f i t maximization or monopoly—but contrad ic tory r esu l t s in the short run fo r the two polar expectat ions assumptions. For a l l behavioral models, in the s t a t i c expectat ions case CU w i l l equal unity between tg and t p with the f i rm on a steady state path. A f t e r the shock at t p the f i rm w i l l no longer be in long run equ i l i b r i um ; CU w i l l exceed un i t y , s ince Y* i s constra ined by the o r i g i na l K l e v e l . This can be viewed as a temporary equ i l ib r ium for the f i rm i s equa l iz ing marginal benef i ts and marginal costs given the f i xed stock -206-l e v e l s , but i t i s not at an equ i l ib r ium in the intertemporal sense. Th is ind i ca tes that incent ives ex i s t f o r the f i rm to invest to increase capac i ty un t i l in the long run CU once again reaches unity with cap i t a l f u l l y ad justed. Hence CU dec l ines monotonical ly to unity a f t e r the shock. In the monopol ist ic case , during the adjustment per iod the output pr i ce as well as the var iab le inputs can be adjusted to c lose the gap between capac i ty and 24 demand in the short run. In the non-stat ic expectat ions case with one permanent ce r t a i n increase in demand, a cont ras t ing adjustment process can be expected in the sense that substant ia l adjustment to the long run equ i l ib r ium leve l may occur during t Q to t p depending on the an t i c ipa ted time in te rva l preceding the shock. This impl ies that the CU measure w i l l drop below unity at t g , corresponding to excess capac i ty in the short run from forward-looking expectat ions , and w i l l equal one sometime a f t e r t^ (say, at t^) when a l l adjustment has been completed and adjustment costs have been "spread" opt imal ly from t Q to t 2 « The adjustment path therefore w i l l be character ized by a monotonic decrease below unity un t i l t^ when a sharp increase occurs to above un i t y , but not to as high a level as in the s t a t i c expectat ions case, for some adjustment has been permitted to take p lace . From there on a monotonic 25 approach to unity w i l l be exh ib i t ed . Th is ana lys i s impl ies that with s t a t i c expectat ions of an expanding economy CU always exceeds unity (as has been observed empi r i ca l l y by BMW), and with non-stat ic and per fec t expectat ions CU always f a l l s short of un i t y . 26 Analogous r esu l t s can be derived fo r q R and product i v i t y changes. Although not pursued here, an important r e su l t i s that both the s t a t i c and non-stat ic cases generate p rocyc l i c a l behavior of q^ and labor product i v i t y with respect to CU, although adjustment in the former case i s a f t e r the shock and i n the l a t t e r case lags behind the shock. -207-Th i s summary of the e f f e c t s of d i f f e r e n t assumptions on the var ious c y c l i c a l ind ica tors demonstrates how genera l i za t ions of the f i rm ' s behavior in rather simple cases w i l l a f f e c t i n t e rp r e t a t i on . The complexi t ies which r esu l t even in these " s t y l i z e d cases" suggest many compl icat ions assoc iated with the der i va t ion and in te rp re ta t i on of i nd i ca to rs of c y c l i c a l behavior of the f i r m . Th is l i n e o f reasoning w i l l be fu r ther explored in the der iva t ion of the mathematical models in the next sec t ion . -208-IV. Issues in Capacity U t i l i z a t i o n Measurement: A More Formal Ana ly t i ca l Treatment In order emp i r i ca l l y to charac te r ize the measures of CU developed in sect ion III, they must be made amenable to ana l y t i ca l representat ion . For tunate ly , ana l y t i ca l cha rac te r i za t i on i s f a i r l y s t ra ight forward , once the CU measure fo r the CRTS cos t funct ion case i s re in terpreted to provide a framework and base case that more r ead i l y extends to the more complex models. Such an i n te rp re ta t i on depends on the not ion , proposed in Berndt-Fuss (1981), of the f i r m ' s shadow cos t func t ion . This shadow cost funct ion i s represented by the to ta l cost funct ion with the con t r ibu t ion of cap i t a l assessed at i t s shadow rather than market va lues. Given G(Y.P.. , t , K , K ) , the shadow cost funct ion can be charac te r ized by: 3.4.1) G (Y ,P . , t ,K , k ) + Z K K, where K i s the "g iven" current leve l of cap i t a l and Z K i s the corresponding 27 shadow va lue . As i n Lau (1976), in the simple var iab le cos t funct ion case where dynamics are not e x p l i c i t l y charac ter ized or where the dynamic model i s constra ined to in terna l costs on net investment, th i s shadow value can be spec i f i ed as -G K « Thus (3.4.1) becomes: 3.4.2) G f Y . P p t . K . K ) - G K K. In a s t a t i c opt imizat ion short run formulat ion K i s not determined with in the model, so that cos ts of adjustment and the r esu l t i ng investment process are 28 not e x p l i c i t l y charac ter ized . By con t ras t , with the dynamic model the cap i t a l path i s determined with in the model. Thus the ex i s t i ng "d i s equ i l i b r i um" in terms of the costs imposed from investment required to c lose the "d i sequ i l i b r ium gap" must be taken into account. -209-With CRTS, the f i rm i s always producing at the minimum of the shadow cos t func t ion , fo r in a sense th i s charac ter izes the long run equ i l ib r ium given the shadow value of c ap i t a l 1^ as the e f f e c t i v e " p r i c e " of K. Thus i f the shadow cos t funct ion i s set equal to the tota l cost func t ion , the point where these two minimum points co inc ide (or where the actual values of output and cap i t a l are cons i s ten t with long run opt imizat ion by the firm) i s determined; there w i l l be no incent ive fo r the f i rm to move from th i s po in t . The equ i l ib r ium cond i t ion fo r the f i rm can be re-stated given th i s i n t e rp r e t a t i on : 3.4.3) G(*) - GKK= G(-) + P RK, r e su l t i ng as above in -G^=P^. Since th i s charac te r izes the s i t ua t i on where K and Y are each at optimal l e ve l s with respect to the leve l of the other , by the d e f i n i t i o n of equ i l ib r ium th i s equation charac te r izes the optimal K* leve l given Y, or the optimal Y* leve l given K. Thus (3.4.3) can be used to "adjust " e i ther of these values to der ive the leve l of the other which i s cons is tent with a steady state or optimum. The shadow cost funct ion revalues the cap i t a l stock so that the current K i s the optimal leve l of K given Y and there i s no incent ive to move from th i s po in t . Thus the shadow cost funct ion in e f f e c t " s h i f t s " the cost curve by i t s reva luat ion of the quas i-f ixed inputs . As K approaches i t s optimal long run va lue , the shadow value approaches the true P R so that K ceases being a cons t ra in t on op t im iza t ion , and the shadow cost funct ion approaches the true SRAC evaluated at the given PK- Imposition of an equ i l ib r ium condi t ion in terms of equating the shadow cost funct ion with the tota l cos t funct ion in e f f e c t forces t h i s equa l i t y . Th is suggests an a l t e rna t i ve motivat ion fo r reva luat ion of the SRAC func t i on , which can a lso be considered a method of s h i f t i n g the f i rm back to where the given cap i t a l stock would have been opt imal . Rather than imposing a -210-tangency, or minimum of SRAC at the given Y, the equ i l ib r ium condi t ion can be * in te rpre ted as s h i f t i n g the e f f e c t i v e production leve l by va lu ing K and K as equ i l i b r ium values and adjust ing Y to i t s corresponding equ i l ib r ium value. Thus, in t h i s sense, the dev ia t ion of Z K from P K , o r , equ i va len t l y , the dev ia t ion of the shadow cost from the to ta l cos t funct ion given current values o f both Y and K, charac te r izes the d i sequ i l i b r ium and can be expressed in terms of the dev ia t ion of Y* from Y, as represented by CU. One can therefore always recover the Y* or s tat ionary point corresponding to the given s i t ua t i on (the given and observed Y, K, and K) , s ince by assumption one i s always on the optimal adjustment path. This i s i m p l i c i t from the fac t that K* i s a funct ion of current Y, hence K i s a funct ion of current Y, so that , f i n a l l y , Y* i s a funct ion of current Y. The c i r c u l a r i t y impl ied i n t h i s d iscuss ion can be unravel led and summarized by noting that with given K and Y and input p r i c e s , the "d i s equ i l i b r i um" can be viewed in a l t e rna t i ve ways. F i r s t , given that K and Y are exogenous and not cons i s ten t with a s ta t ionary state at the given p r i c e s , c a l c u l a t i o n of a shadow cost funct ion w i l l revalue these pr ices so that given these a l t e rna t i ve pr i ces one i s at a s ta t ionary s ta te . Th is approach charac te r izes the d i sequ i l i b r ium in terms of the va r i a t ion between and P^. If one equates th i s funct ion with the LRAC func t i on , and thus in e f f e c t forces the equa l i t y of and P ,^ something e lse has to " g i v e " . Given the cur rent values of K and Y th i s cannot ho ld , but i t can hold i f K i s stated in terms of the given Y (der ivat ion of K*) or i f Y i s stated in terms of the given K (der iva t ion of Y*) . Pursu i t of e i the r of these approaches imposes consistency of the K and Y values (and thus a s tat ionary state) given a l l other values of the exogenous va r i ab l es . This approach therefore impl ies two a l t e rna t i ve but equiva lent cha rac te r i za t ions of the observed d i sequ i l i b r ium in the economy, K*-K (or , K*/K), or Y*/Y, which I have a l t e rna t i ve l y defined as -211-th e "gap" between des i red K* and given K that i s to be c losed by investment, and the measure of CU, r espec t i ve l y . Thus i n some sense the cha rac te r i za t i on of Y* given K or K* given Y are "dua l " r e l a t i o n s h i p s , and th i s " dua l i t y " of the c a p i t a l - and output-oriented approaches to cha rac te r i za t ion of the optimum can a l t e rna t i v e l y be represented by the K- and Y-oriented approaches to measuring the d i s equ i l i b r i um , Tob in ' s q and CU, r e spec t i v e l y . The re l a t ionsh ip can be i l l u s t r a t e d fur ther by noting that in equ i l ib r ium both of these measures are equal to un i t y . However, in d i sequ i l i b r i um th i s i s not the case , and the extent of d i sequ i l i b r ium can be measured by the magnitude of the dev ia t ion of q and/or CU from un i ty . Th is "dua l i t y " concept emphasizes the l inkage between the shadow value of cap i t a l and the Y* measure and thus points out the importance of proper ly cha rac te r i z ing the shadow value of c ap i t a l in order to ca l cu l a t e CU as well as q R . S p e c i f i c a l l y , i t i s important to adjust the stock or iented shadow value o f c ap i t a l from simpler models fo r the amortized adjustment costs represented by r G R . Such a modi f i ca t ion r e l i e s on the e x p l i c i t cha rac te r i za t ion of the "f low" as contrasted to the "s tock" aspects of the f u l l dynamic problem. The existence of rG^ r e f l e c t s the essent ia l dynamic nature of the problem even in equ i l i b r i um, as suggested by the general equ i l ib r ium cond i t ion from Essay 2: 3.4.4) -(G K+rG K ) = PK-With costs of adjustment on net investment and an equ i l ib r ium framework G R (0)=0. Thus th i s term drops out of the equ i l ib r ium charac te r iza t ion i f de r i va t ion i s based on s t a t i c opt imizat ion ana lys i s def ined in terms of "g iven" K and K va lues , although in d i sequ i l i b r ium K#) so G RtK)^0 in general and should be taken into account i n der i va t ion of q and CU. Thus, the compensating adjustmentrG£ can be viewed as a wedge resu l t i ng from costs of 29 adjustment and the corresponding dynamic adjustment process. -212-This d i s t i n c t i o n between stock and flow aspects can be in terpreted in terms of Keynes' d i s t i n c t i o n between the marginal e f f i c i enc y of cap i t a l and the marginal e f f i c i e n c y of investment, as I have a l luded to above. The term rG^ can be in te rpre ted as a determinant of the marginal e f f i c i e n c y of investment in cont ras t to the marginal e f f i c i e n c y of c a p i t a l , the l a t t e r charac ter ized by a lone. -G^ can be in terpreted as the marginal value o f an incremental un i t o f c a p i t a l , or the marginal value of obta in ing one more un i t of c ap i t a l but re ta in ing the same investment l e v e l . In t h i s case , two states are being compared with investment f i x e d . In a sense th i s i s based on a comparative s t a t i c s framework as i t only takes stock l eve l s in to account. If i t requi res investment to reach th i s new cap i t a l l e v e l , however, the r esu l t i ng adjustment cos t in terms of G£ must be included e x p l i c i t l y . Thus even along the steady state path , the true cos t of a uni t of cap i t a l i s equal to the un i t p r i ce of cap i t a l p lus amortized adjustment cos ts—the to ta l marginal cost of c a p i t a l . The rG^ term i s thus an important component of the shadow value of K, and p a r t i c u l a r l y of q^, as q^ i s by const ruc t ion a 30 determinant of investment behavior. The proper q^ measure i s therefore represented by -(G^+rGj^/P^, and Y* can be charac ter ized by imposing the equa l i ty of -(G^+rGj^) and P^ and f i nd ing the impl ied Y*. Note that t h i s extension can be in terpre ted in two d i f f e r e n t but a n a l y t i c a l l y equiva lent ways by i n se r t i ng rG^ a l t e rna t i v e l y on the l e f t or r i gh t hand side of the equa l i t y spec i f i ed in (3 .4 .3 ) . I.e., fo r the c a l c u l a t i o n of the shadow cost and to ta l cost equa l i ty cond i t i on , I can i n t e rp re t -G K as the shadow rental value of the cap i t a l stock, and therefore inc lude only t h i s term in the shadow value func t ion . In t h i s case I can subst i tu te P K+rG^ fo r the P^ in the to ta l cost func t i on , which in e f f e c t augments the renta l p r i ce of K fo r the amortized costs of adjustment incurred -213-at the given stock l e v e l . This r e su l t s in an equ i l ib r ium cond i t ion for the f u l l model as : 3.4.25) - G K = P K + r G R , which can be in te rpre ted as a stock va luat ion expression i f one envisages - G K in terms of the marginal e f f i c i e n c y o f c ap i t a l concept. The a l t e rna t i ve equiva lent i n te rp re ta t i on i s , however, more useful i f one were to use the framework to const ruct a measure of Tob in ' s q, as an ind i ca to r of investment determinat ion. For the q measure one would want to use the expression represented by (3 .4 .4 ) , where the l e f t hand s ide i s in te rpre ted as the marginal e f f i c i e n c y o f investment; the shadow value of cap i t a l inc ludes here the cos t e f f e c t s of the cap i t a l adjustment, not j u s t the marginal eva lua t ion . Note, however, that these two approaches are a n a l y t i c a l l y equivalent--the amortized adjustment costs are simply t rans fe r red to a d i f f e r e n t side of the equa l i t y . It i s only the i n te rp re ta t i on that i s a l t e red to correspond to the d i f f e r e n t purposes. Now cons ider the case when adjustment costs are spec i f i ed in terms of gross ( rather than net) investment, and where external as well as in te rna l costs are inc luded. As discussed in Essay 1, th i s impl ies fur ther complexity of the shadow value of c a p i t a l . The f i r s t point to note in th i s case i s that the conventional equ i l ib r ium expression -GR+rG^=PR i s a lso v a l i d in t h i s case . This can be shown in an i n t u i t i v e manner by not ing that the ca l cu lus o f va r i a t ions problem i s expressed in terms of K and K instead of K and Z. Thus, i f G i s wr i t ten by subs t i t u t i ng Z=k+dK and i t i s recognized that the replacement component of gross investment, dK, w i l l s t i l l be non-zero in long run equ i l i b r i um , i t i s easy to der ive that the above equ i l i b r ium expression i s equiva lent to : -214-3.4.6) - G K + (r+d)G z = P K , the investment path equ i l ib r ium equa l i t y in terms of Z instead of K. Th is corresponds bet ter to i n t u i t i o n as i t more c l o se l y resembles the equ i l i b r ium expressions from t r ad i t i ona l neoc lass i ca l theory. Along the optimal investment path, the rental pr ice of cap i t a l can therefore be in te rpre ted as the marginal c ap i t a l stock evaluat ion inc lud ing the required replacement investment (-GK) p lus the marginal adjustment costs of new investment, (-rG£ or - ( r+d)G z ) , r e f l e c t i n g the impact of deprec iat ion and adjustment cos t s . In tu rn , the impl ied q value and re la ted Y measure must a lso incorporate t h i s extra deprec ia t ion c o s t , "the marginal adjustment cos t of i n s t a l l i n g the annual replacement investment required to maintain the des i red 31 cap i t a l stock" . Incorporat ion o f external as well as in terna l cos ts o f adjustment as was done in Essay 1 adds more expressions to the shadow value of c a p i t a l . S p e c i f i c a l l y , in th i s case the to ta l cost funct ion may be wr i t ten as G+P K K+a K S (K ,£ ) , where a K i s the asset p r i ce of investment and S(K,k) i s the external costs of cap i t a l func t ion , represented in Essay 1 by . 5 d ^ Z . Total costs therefore depend on both the technology the f irm faces , charac te r ized by the var iab le cost fun t ion , and the exogenous market forces in the cap i t a l goods market over which the f i rm must optimize represented by the external adjustment costs func t ion . Given th i s extended cost func t i on , the equ i l i b r ium cond i t ion must take external adjustment cos t funct ion components as well as var iab le cost funct ion components into account. This was der ived in Essay 1, reproduced here in a form more useful for the current a p p l i c a t i o n : 3.4.7) - ( (G K + a R S K ) + ( r (G£ + a K S-) ) )=P K . -215-Here the shadow value of cap i t a l in terms of investment i s represented by the en t i r e l e f t hand s ide of t h i s express ion, o r , Z K=-(G R+rG^+a KS K+ra KS^). Th is must be recognized in the der i va t ion of q R and shadow cos t funct ion or CU measures. Given these shadow value expressions for the var ious models, I now return to i n t e rp re t a t i on of the o r i g i na l CRTS cos t funct ion measures of CU, using the net investment model in the shadow cost functon framework. I then proceed to ana lys i s of the more complex cases . The graphical i l l u s t r a t i o n of the simple case of Y* determination i s presented in F igure 1 1 1 - 9 . Given the L R A C curve L R A C Q , say that the f i rm i s at a " d i s e q u i l i b r i u m " pos i t i on with short run increas ing un i t cos ts charac te r ized by S R A C Q ( K Q . K Q ) and Y j . Y^, perhaps as a consequence o f a previous unexpected increase in exogenous output demand, i s l a rger than the production leve l corresponding to the minimum point on the current S R A C Q 33 curve defined by K Q . Respeci fy ing the to ta l cost curve in terms of shadow pr i ces or shadow values fo r the given K , K and Y l eve l s re locates the " e f f e c t i v e " curve at SRAC g n . Uni t costs are at a higher leve l s ince the shadow value of cap i t a l exceeds the e x i s t i n g market p r i c e ; cap i t a l i s a binding cons t ra in t to opt imizat ion and more cap i t a l i s des i red to move to S R A C * ( K * ) . F igure 9 demonstrates g raph i ca l l y the " dua l i t y " of the d i sequ i l i b r ium measures, q K and u d iscussed e a r l i e r . S R A C Q (corresponding to the to ta l cost curve given K and e x i s t i n g pr ices ) d i f f e r s from SRAC S^ (corresponding to the cos t curve with K revalued at the shadow pr i ces so that the minimum of 34 the curve occurs at the e x i s t i n g Y l e v e l ) . I f e x i s t i n g and shadow p r i c e s were equal the two curves would co inc ide at a s tat ionary point with K Q and Y p and q^ would be equal to un i t y . A l t e r n a t i v e l y , i f one imposes the equ i l ib r ium cond i t ion in terms of the equa l i t y of the shadow values and the given long run va lues , and th i s i s not cons is tent with the exogenously given - 2 1 6 -Figure II1-9 Figure I11-10 -217-Kg, Y p e i the r K or Y must "g i ve " fo r the equa l i t y to ho ld . I f K were allowed to adjust to the impl ied value cons is tent with a steady s ta te , K* would be determined simply by dropping the SRAC S H curve d i r e c t l y down to a tangency with L R A C Q at Y ^ This in turn def ines SRAC*(K*). I f , however, K were held f ixed and output were adjusted to a
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Three essays in the dynamic analysis of demand for factors of production Morrison, Catherine J. 1982
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Title | Three essays in the dynamic analysis of demand for factors of production |
Creator |
Morrison, Catherine J. |
Publisher | University of British Columbia |
Date Issued | 1982 |
Description | This dissertation consists of three essays focussing on various theoretical extensions and empirical implementations of a model characterizing the dynamic input demand behavior of the firm. The analysis is based on recent developments on optimal investment decisions over time subject to increasing marginal costs of adjustment. Theoretical extensions to the model include the incorporation of (i) monopoly behavior based on profit maximization with both internal and external costs of adjustment on gross investment (Essay 1) and (ii) non-static output and price expectations (Essay 2). Both theoretical models are empirically implemented using annual U.S. manufacturing data, 1948-77. Results from the various specifications can therefore be analyzed and contrasted. Essay 3 consists of an application of the alternative models to measures of capacity utilization specified in terms of the explicit dynamic optimization model of the firm's behavior. Various derivations and interpretations of these measures for alternative theoretical specifications are developed and compared, both analytically and empirically. The three essays are linked by a common structural model which provides a basis for incorporation of different behavioral assumptions, and for examination and comparison of the various models in terms of demand elasticities, cyclical indicators such as multifactor productivity and Tobin's "q", and capacity utilization. |
Subject |
Demand (Economic theory) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-02 |
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 | 10.14288/1.0095899 |
URI | http://hdl.handle.net/2429/24332 |
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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