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Two models of dynamic input demand : estimates with Canadian manufacturing data 1990

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TWO MODELS OF DYNAMIC INPUT DEMAND: ESTIMATES WITH CANADIAN MANUFACTURING DATA By MICHAEL JOHNSTONE RUSHTON B.A., The Un i v e r s i t y of B r i t i s h Columbia, 1 9 8 0 M.A., The Un i v e r s i t y of Western Ontario, 1 9 8 1 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Economics We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1 9 9 0 © Michael Johnstone Rushton, 1 9 9 0 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of bcovNcrw^ics The University of British Columbia Vancouver, Canada Date o<-Ao^>er 3> , DE-6 (2/88) ABSTRACT Over the past decade there has been a number of innovations i n the estimation of input demand equations. In p a r t i c u l a r , ways of incorporating the hypothesis of r a t i o n a l expectations into empirical models of the f i r m have been developed and improved upon. This research agenda was perhaps i n s p i r e d by the Lucas c r i t i q u e of econometric p o l i c y evaluation, which suggested that econometric models which d i d not e x p l i c i t l y take account of how expectations of the future a f f e c t current behaviour would give misleading r e s u l t s regarding the possible e f f e c t s of various government p o l i c i e s . Lucas s p e c i f i c a l l y d i r e c t e d part of his c r i t i q u e at empirical models of business investment, which had been used previously i n the assessment of tax p o l i c i e s designed to a f f e c t investment. This t h e s i s has a dual purpose. F i r s t , two d i s t i n c t models of input demand are estimated with Canadian manufacturing data. Each of the models incorporates to some degree the hypothesis of r a t i o n a l expectations, but the s p e c i f i c a t i o n s of technology d i f f e r . Neither of these models, to our knowledge, has been estimated with Canadian data. We are int e r e s t e d i n whether e i t h e r model explains well the behaviour of the Canadian manufacturing sector, and i n how the r e s u l t s compare with the (few) U.S. applications of t h i s type of model. The second purpose i s to use the r e s u l t s of these models i n simulations to assess the e f f e c t of changes to the aft e r - t a x i i r e n t a l rate of c a p i t a l on investment and employment i n manufacturing. While there have been studies i n Canada (and elsewhere) that attempt to c a l c u l a t e the e f f e c t s of various tax p o l i c i e s on investment, most studies were done p r i o r to the innovation of techniques i n estimating models with r a t i o n a l expectations. This t h e s i s i s able to examine the e f f e c t s of a p a r t i c u l a r change while remaining immune to the Lucas c r i t i q u e . If the modelling of expectations i s correct, t h i s could not only improve the r e l i a b i l i t y of the estimates, but also give some i n d i c a t i o n of the empirical importance of the Lucas c r i t i q u e . The r e s u l t s can be summarized as follows. The two models give very d i f f e r e n t estimates of p r i c e e l a s t i c i t i e s of demand for c a p i t a l and labour, even though they are s i m i l a r i n many respects and are estimated with a common data set. It i s also the case that t h e i r estimates of the e f f e c t s of temporary and permanent changes to the r e n t a l rate are d i f f e r e n t . Adjusting the reduced form parameters of the input demand equations to account for changes i n tax p o l i c y regimes a l t e r s the r e s u l t s to a s i g n i f i c a n t degree, suggesting that the e x p l i c i t modelling of expectations matters i n an e m p i r i c a l l y relevant sense. However, these e f f e c t s are i n opposite d i r e c t i o n s for the two models considered here. A l l t h i s suggests that more research i s required into the r e l a t i o n s h i p between expectations of future p o l i c y and investment behaviour. i i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v ACKNOWLEDGEMENT S ' v i CHAPTER 1 - INTRODUCTION 1 1.1 The Neo c l a s s i c a l Model of Investment with D i s t r i b u t e d Lags 2 1.2 The Neoclassical Model of Investment with Adjustment Costs 6 1.3 Rational Expectations and the Lucas C r i t i q u e 9 1.4 Modelling Investment with Rational Expectations 14 1.5 Tax P o l i c y and Investment 20 CHAPTER 2 - THE DATA 30 2.1 Wages and Rental Rates 30 2.2 C a p i t a l and Labour Inputs 34 2.3 Output 35 CHAPTER 3 - THE LINEAR QUADRATIC MODEL 37 3.1 A Time Series Analysis of the Data 37 3.2 The Model 47 3.3 Estimates 53 3.4 Simulations 64 3.5 Conclusions 68 CHAPTER 4 - EPSTEIN AND DENNY'S MODEL ....75 4.1 The Model 77 4.2 Estimates 81 4.3 The Model with Non-Static Expectations 88 4.4 Simulations with Non-Static Expectations 96 4.5 Comparing the Performance of the Models 99 4.6 Conclusions 103 CHAPTER 5 - CONCLUSION 108 REFERENCES 112 i v LIST OF TABLES Table Page 2.1 Data 36 3.1 Estimate of u n r e s t r i c t e d linear-quadratic model where input demands depend on current input prices 70 3.2 Estimate of u n r e s t r i c t e d linear-quadratic model where input demands depend on lagged input prices 71 3.3 Estimate of r e s t r i c t e d linear-quadratic model where input demands depend on current input prices 72 3.4 Estimate of r e s t r i c t e d linear-quadratic model where input demands depend on lagged input prices 73 3.5 Forecast values of the c a p i t a l stock under various conditions 7 4 4.1 R e s t r i c t e d estimates of f l e x i b l e f u n c t i o n a l form model with s t a t i c expectations 105 4.2 R e s t r i c t e d estimates of f l e x i b l e f u n c t i o n a l form model with autoregressive expectations 106 4.3 Forecast values of the c a p i t a l stock under various conditions, using the f l e x i b l e f u n c t i o n a l form model 107 v ACKNOWLEDGEMENTS I am gr e a t l y indebted to my supervisory committee of Professors John Cragg, John H e l l i w e l l (supervisor), and Jeremy Rudin f o r t h e i r c r i t i c i s m s , suggestions, and patience. I also thank Professors Ardo Hansson, Chris N i c o l , William Schworm, Michael Tretheway, and Tom Wilson f o r t h e i r suggestions. I am responsible f o r the errors that remain. The members of the economics departments at the u n i v e r s i t i e s of Tasmania and Regina provided support and encouragement while I f i n i s h e d w r i t i n g the t h e s i s . F i n a l l y , my thanks and my love go to my family, who always believed i n me no matter how many d i f f e r e n t times I would t e l l them I was "jus t about done". v i 1 CHAPTER 1 Introduction Over the past decade there has been a number of innovations i n the estimation of input demand equations. In p a r t i c u l a r , ways of incorporating the hypothesis of r a t i o n a l expectations into empirical models of the f i r m have been developed and improved upon. This research agenda was perhaps i n s p i r e d by the Lucas (1976) c r i t i q u e of econometric p o l i c y evaluation, which suggested that econometric models which d i d not e x p l i c i t l y take account of how expectations of the future a f f e c t current behaviour would give misleading r e s u l t s regarding the possible e f f e c t s of various government p o l i c i e s . Lucas s p e c i f i c a l l y d i r e c t e d part of h i s c r i t i q u e at empirical models of business investment, which had been used previously i n the assessment of tax p o l i c i e s designed to a f f e c t investment. This t h e s i s has a dual purpose. F i r s t , two d i s t i n c t models of input demand are estimated with Canadian manufacturing data. Each of the models incorporates to some degree the hypothesis of r a t i o n a l expectations, but the s p e c i f i c a t i o n s of technology d i f f e r . Neither of these models, to our knowledge, has been estimated with Canadian data. We are int e r e s t e d i n whether e i t h e r model explains well the behaviour of the Canadian manufacturing sector, and i n how the r e s u l t s compare with the (few) U.S. appl i c a t i o n s of t h i s type of model. The second purpose i s to use the r e s u l t s of these models to 2 assess the e f f e c t of the r e n t a l rate of c a p i t a l i n Canada on investment and employment i n manufacturing. While there have been several studies i n Canada (and elsewhere) that attempt to c a l c u l a t e the e f f e c t s of various tax p o l i c i e s on investment, most studies were done p r i o r to the innovation of techniques i n estimating models with r a t i o n a l expectations. This t h e s i s w i l l be able to examine the e f f e c t s of temporary and permanent changes i n the r e n t a l rate while remaining immune to the Lucas c r i t i q u e . If the modelling of the expectations process i s correct, t h i s could not only improve the r e l i a b i l i t y of the estimates, but also give some i n d i c a t i o n of the empirical importance of the Lucas c r i t i q u e . In the remainder of t h i s chapter, sections 1.1 through 1.4 give a b r i e f h i s t o r y of modelling input demands and investment, and sketch out the o r i g i n s of the models estimated i n t h i s t h e s i s . Section 1.5 contains a short survey of research on the question of the e f f e c t s of tax p o l i c y on investment, and shows how the method of p o l i c y analysis we have adopted d i f f e r s from those used i n previous studies. 1.1 The N e o c l a s s i c a l Model of Investment with D i s t r i b u t e d Lags The essence of n e o c l a s s i c a l theories of investment i s that a f i r m w i l l choose i t s path of c a p i t a l accumulation, and of other inputs, with the objective of maximizing the present discounted value of the income stream generated by the firm. Key secondary assumptions (see Jorgenson (1967, p.136)) are that the f i r m purchases a l l inputs, i n c l u d i n g c a p i t a l goods, i n competitive 3 markets, that the fir m s e l l s output i n a competitive market, and that the contribution of c a p i t a l to output i s through a flow of services proportional to the stock of c a p i t a l . It can be presumed that the f i r m purchases c a p i t a l goods to use i t s e l f , or that the f i r m leases c a p i t a l goods from a separate owner. In e i t h e r case, the cost of using a unit of c a p i t a l f o r one time period, c a l l e d the user cost of c a p i t a l or the r e n t a l rate of c a p i t a l , i s the same, and i s a key determinant of the optimal c a p i t a l stock f o r the firm. Standard microeconomic theory of the fi r m suggests that, i f the marginal product of c a p i t a l i s decreasing with the l e v e l of c a p i t a l input, and i f the fir m may obtain any c a p i t a l stock i t wishes at market p r i c e s , firms w i l l desire a l e v e l of c a p i t a l input such that the marginal product of c a p i t a l i s equal to the user cost. In the absence of corporate taxes or subsidies, the user cost of one d o l l a r ' s worth of c a p i t a l f o r one year i s equal to the in t e r e s t rate plus the depreciation rate on c a p i t a l minus any c a p i t a l gains r e s u l t i n g from a change i n the p r i c e of c a p i t a l goods over the year (see Jorgenson (1967, p.143) or Boadway (1980, p.253)) . Corporate taxation, together with tax exemptions f or depreciation and i n t e r e s t , y i e l d s a more complicated expression f o r the user cost of c a p i t a l ; t h i s i s described l a t e r i n Chapter 2. It has long been accepted that while the n e o c l a s s i c a l model of the f i r m can give us a theory of the demand for c a p i t a l , t h i s i s not the same thing as a theory of investment. Consider an often c i t e d extract from Haavelmo ( 1 9 6 0 , p. 2 1 6 ) : What we should r e j e c t i s the naive reasoning that there i s a demand schedule f o r investment which could be derived from a c l a s s i c a l scheme of producers' behavior i n maximizing p r o f i t . The demand f o r investment cannot simply be derived from the demand for capital. Demand for a f i n i t e addition to the stock of c a p i t a l can lead to any rate of investment, from almost zero to i n f i n i t y , depending on the a d d i t i o n a l hypothesis we introduce regarding the speed of reaction of ca p i t a l - u s e r s . I think that the sooner t h i s naive, and unfounded, theory of the demand-for-investment schedule i s abandoned, the sooner we s h a l l have a chance of making some r e a l progress i n constructing more powerful theories to deal with the capricious short-run v a r i a t i o n s i n the rate of private investment. (Emphasis i n o r i g i n a l ) . This view has not been u n i v e r s a l l y accepted however. B l i s s ( 1 9 7 5 , p. 304) points out that i f we have determined the demand fo r c a p i t a l at time t and at time t+ 1 then we have also determined the rate of net investment over that time i n t e r v a l . What Haavelmo seems to be saying, suggests B l i s s , i s that the demand f o r net investment w i l l not ne c e s s a r i l y be equal to the change i n the le v e l s of c a p i t a l stock that equates marginal product to user cost from one period to the next, because f o r various reasons (described below) firms w i l l not always choose those l e v e l s of c a p i t a l . In any case, Haavelmo, when discussing the 'demand f o r 5 investment' i s r e f e r r i n g to what firms a c t u a l l y do, whereas when discussing the 'demand f o r c a p i t a l ' he i s r e f e r r i n g to the l e v e l s of c a p i t a l firms would choose i n a world of no time lags on the de l i v e r y of c a p i t a l goods and no adjustment costs with respect to the l e v e l s of c a p i t a l input. Jorgenson's s o l u t i o n to the problem posed by Haavelmo i s as follows. Let K be the l e v e l of the c a p i t a l stock at time t that s a t i s f i e s the condition of marginal product of c a p i t a l equals the user cost. Assume that each period new investment projects are i n i t i a t e d such that the backlog of uncompleted projects i s equal * to the dif f e r e n c e between and the actual stock of c a p i t a l K . Let P/(L) be a power serie s i n the la g operator L , which describes the rate at which investment projects i n progress become completed, and which i s given. I f 5 i s the depreciation rate, and gross investment i n any period equals investment i n new projects plus replacement investment, then gross investment at time t i s given by I f c = w(L) [K* - K*_1] + 5 K f c ( 1 . 1 ) (Jorgenson ( 1 9 6 3 , p p . 2 4 9 - 5 1 ) ) . The demand f o r c a p i t a l i s thus transformed into a 'demand for investment' by the a p p l i c a t i o n of a r a t i o n a l d i s t r i b u t e d l a g process (see Jorgenson ( 1966) for an analysis of the properties of such a process). Jorgenson ( 1 9 6 3 ) and H a l l and Jorgenson ( 1 9 6 7 ) estimate the parameters of investment functions l i k e ( 1 . 1 ) , and from these are 6 able to i n f e r the response of investment with respect to changes i n user costs and output p r i c e s . Eisner and N a d i r i (1968) are c r i t i c a l of H a l l and Jorgenson's reliance on the Cobb-Douglas s p e c i f i c a t i o n of technology, when the evidence suggests that something more general i s c a l l e d for, and of t h e i r method of choosing an appropriate lag structure, since i t turns out that r e s u l t s are highly dependent on the r e s t r i c t i o n s placed on w(L). Gaudet, May, and McFetridge (1976) apply the n e o c l a s s i c a l model with a d i s t r i b u t e d l a g to Canadian manufacturing data (1952 to 1 9 7 3 ) , using a CES production function (which i s recommended by Eisner and Nadiri) . They suggest that the r e s u l t s of the model could be useful i n the analysis of tax p o l i c i e s , since they f i n d that the user cost of c a p i t a l has a s t a t i s t i c a l l y s i g n i f i c a n t e f f e c t on investment. How the modelling of investment has changed from the method of Jorgenson over the past two decades i s the subject of the next three sections. 1.2 The Ne o c l a s s i c a l Model of Investment with Adjustment Costs The concept of adjustment costs was incorporated into the ne o c l a s s i c a l theory of investment because, i n the words of Lucas (1967b, p.78) : ...many students of investment behavior have recognized the incongruity of developing a rigorous economic theory of the determination of [K^] and then combining t h i s with an ad hoc theory of adjustment. 7 When we speak of adjustment costs we mean that i n the production function output depends not only upon the l e v e l s of inputs, but also on the rate at which these l e v e l s are changing. In p a r t i c u l a r , the f a s t e r the l e v e l s of inputs are changing, the lower w i l l be the rate of output, ceteris paribus. Gould (1968), Lucas (1967a, 1967b), Mortensen (1973), Schramm (1970), and Treadway (1969, 1970, 1971) a l l specify adjustment costs as convex. That i s , the cost to the f i r m of changing i t s l e v e l of c a p i t a l input i n any period i s increasing with the absolute value of the change and i s increasing at an increasing rate. A l l of the studies mentioned above base adjustment costs on the rate of net investment, with the exception of Gould, who uses gross investment, and Lucas (1967a), who uses gross investment as a proportion of c a p i t a l stock. Since i t i s assumed i n these studies that c a p i t a l markets are competitive, the ration a l e given f o r adjustment costs i s that when there i s net investment output i s le s s , given the stocks of c a p i t a l and labour, than i t would be i n the absence of net investment. This i s because some labour and c a p i t a l must be devoted to i n s t a l l i n g , and learning how to work with, the new le v e l s of inputs (see N i c k e l l (1978, Chapter 3)). With adjustment costs incorporated into the n e o c l a s s i c a l model - i . e . i t remains the case that firms seek to maximize the present discounted value of cash flow, and that they buy inputs and s e l l outputs i n competitive markets - the r e s u l t i n g demand f o r net investment has the same form as was obtained by Jorgenson. 8 That i s , net investment i s proportional to the difference between the 'target stock' of c a p i t a l Kfc and actual c a p i t a l K . This reduced form i s c a l l e d the f l e x i b l e accelerator. The differ e n c e between t h i s r e s u l t and Jorgenson's i s that the adjustment cost model gives a net investment function that i s the r e s u l t of equilibrium c a p i t a l accumulation, whereas Jorgenson's i s e s s e n t i a l l y disequilibrium (see B l i s s (1975, p. 305)). In the * adjustment cost model the movement towards K i s not immediate because i t would not be p r o f i t maximizing to do so. In Jorgenson's * d i s t r i b u t e d l a g model movement towards K i s not immediate because there are time lags involved i n completing c a p i t a l projects that cannot be avoided no matter what p r i c e the f i r m i s w i l l i n g to 1 pay. In e i t h e r the d i s t r i b u t e d l ag or the adjustment cost model information about future p r i c e s i s valuable to the firm. I f the e x i s t i n g l e v e l of c a p i t a l input at any time constrains the fir m i n i t s problem of maximizing the present discounted value of future cash flow, then the choice of net investment i n the current period w i l l determine the constraint at the beginning of the next period. If p r i c e s are changing i n some way over time, the way they are See Almon (1965) f o r an empirical study of such unavoidable time lags between c a p i t a l appropriations and expenditures. The notion of unavoidable delays between i n i t i a t i n g a c a p i t a l project and the c a p i t a l being a v a i l a b l e f o r production services reappears i n the "time-to-build" model of Kydland and Prescott (1982). 9 changing w i l l a f f e c t future target c a p i t a l stocks and therefore current net investment. The innovation to empirical models of investment which followed the introduction of adjustment costs was the introduction of r a t i o n a l expectations. 1.3 Rational Expectations and the Lucas C r i t i q u e Rational expectations as c u r r e n t l y applied to economic models dates from Muth (1961). He describes the idea as follows: ...expectations of firms (or, more generally, the subjective p r o b a b i l i t y d i s t r i b u t i o n of outcomes) tend to be d i s t r i b u t e d , f or the same information set, about the p r e d i c t i o n of the theory (or the "objective" p r o b a b i l i t y d i s t r i b u t i o n s of outcomes). In p r a c t i c e , the information set postulated by modellers includes (1) the structure of the model i t s e l f , and (2) a l l past values of relevant v a r i a b l e s . An i m p l i c a t i o n of the assumption of r a t i o n a l expectations i s that forecast errors of firms should be random and uncorrelated to any variables i n the information set or to past values of errors, since any such c o r r e l a t i o n would be useful information to the f i r m that should be incorporated into the subjective p r o b a b i l i t y d i s t r i b u t i o n of outcomes. This i s why r a t i o n a l expectations r e a l l y amounts to a consistency condition; except for a random error term the expectations of firms should be consistent with the outcomes of the model which incorporates the firms. We have mentioned e a r l i e r that corporate tax p o l i c y influences the user cost of c a p i t a l , and that i n the presence of adjustment costs firms w i l l want to make forecasts of user costs to help determine the optimal l e v e l of net investment f o r the current period. This means that under r a t i o n a l expectations the re l a t i o n s h i p between the rate of net investment and the current user cost of c a p i t a l w i l l vary according to expectations of future user costs. Yet H a l l and Jorgenson (1967) t r e a t the r e l a t i o n between investment and user costs as though i t were stable. The method of H a l l and Jorgenson f o r estimating the e f f e c t s of tax p o l i c y on investment was as follows. F i r s t , derive a r e l a t i o n between the user cost and the optimal c a p i t a l stock from * a s t a t i c perspective, K^. Then estimate the d i s t r i b u t e d l ag that * * rela t e s investment to lagged values of K . Then ask how K would have been d i f f e r e n t had user costs been d i f f e r e n t , say because of some hypothetical a l t e r n a t i v e tax p o l i c y . Then, given the distributed lag function which was estimated using actual data, ask how the path of investment would have been d i f f e r e n t . The c r i t i q u e of econometric p o l i c y evaluation by Lucas (1976) i s based on the idea that reduced form parameters of an econometric model, say the d i s t r i b u t e d l ag function w(L) f o r example, w i l l not be invariant to changes i n p o l i c y . Thus p o l i c y analysts should d i s t i n g u i s h between s t r u c t u r a l and reduced form parameters of models (a warning made as ea r l y as 1953 by Marschak). S t r u c t u r a l parameters, f o r the purposes of p o l i c y analysis, are those which are invariant to changes i n the p o l i c y regime, where by p o l i c y regime we mean the process which guides year to year changes to the tax structure. Since i n a model of investment where expectations of future r e n t a l rates matter the reduced form of the investment equation w i l l change with changes i n the p o l i c y regime, the r e s u l t s of H a l l and Jorgenson on the e f f e c t s of tax p o l i c y on investment are not r e l i a b l e . The research agenda suggested by the Lucas c r i t i q u e was to devise means of estimating models which included r a t i o n a l expectations and which could i d e n t i f y the s t r u c t u r a l parameters (in models of the f i r m these are usually taken to be the technology and adjustment cost parameters). With the s t r u c t u r a l parameters estimated the modeller could determine how reduced form parameters would change f or d i f f e r e n t p o l i c y regimes, and could more accurately assess the impacts of various p o l i c i e s . Although much of t h i s research was focussed on the estimation 2 of complete macroeconomic models , we d i r e c t our attention to techniques of estimating models of input demand i n the presence of adjustment costs, c a l l e d dynamic models of input demand. A f i r m which seeks to maximize the discounted stream of returns from production subject to the constraints of e x i s t i n g l e v e l s of inputs, adjustment costs to changing these l e v e l s of inputs, and market p r i c e s f o r inputs and output which evolve over time i n a way beyond i t s con t r o l , i s faced with a problem of See Chow (1983, Chapter 11), Taylor (1979), or Wallis (1980) f o r example. 12 3 optimal c o n t r o l . The f i r s t order conditions f o r the so l u t i o n to the firm's maximization problem include the Euler equations and the t r a n s v e r s a l i t y conditions. The technique r e f e r r e d to as "limited-information estimation" involves estimating the parameters of the model by estimating the Euler equations d i r e c t l y . Kennan ( 1 9 7 9 ) describes how the Euler equations might be 4 estimated e f f i c i e n t l y . Applications of t h i s method to dynamic models of input demand are described i n the following section. Under the technique known as " f u l l - i n f o r m a t i o n estimation" the model i s solved f o r a l l the f i r s t - o r d e r conditions and then estimated. There are two methods of achieving t h i s . Hansen and Sargent ( 1 9 8 0 , 1 9 8 1 , 1982) solve f o r the firm's optimal decision rule f o r input demands i n the current period as a function of lagged values of input l e v e l s and future expected p r i c e s . They then use r e s u l t s of p r e d i c t i o n theory to express future expected pr i c e s i n terms of current information (past observations of variables useful i n forecasting relevant prices and knowledge of the model which determines these v a r i a b l e s ) . This gives an input demand equation which can be estimated since the right-hand side variables - lagged input l e v e l s and the current information set - are a l l observed v a r i a b l e s . Technology parameters of the fi r m and A standard reference i s Chow ( 1975) . For a treatment which incorporates into the optimal c o n t r o l problem recent developments i n r a t i o n a l expectations see Sargent ( 1 9 8 7 , Chapter 1 ) . 4 Also see Hansen ( 1 9 8 2 ) and Hansen and Singleton ( 1982) . 13 parameters which are used i n the pr o j e c t i o n of current information into expected future prices are estimated d i r e c t l y . Simultaneously, the model used to forecast future prices i s also estimated. There w i l l be cross-equation r e s t r i c t i o n s i n t h i s simultaneous estimation, and tes t s of the cross-equation r e s t r i c t i o n s amount to a j o i n t t e s t of the model of the fir m and of the hypothesis of r a t i o n a l expectations. Lucas and Sargent (1981, p. x v i i ) r e f e r to the cross-equation r e s t r i c t i o n s as a "hallmark" of r a t i o n a l expectations models; the firm's decision rule f o r input demand i s e x p l i c i t l y r e l a t e d to the model used i n forecasting p r i c e s . Chow (1980a, 1981) takes a somewhat d i f f e r e n t approach, although the estimation remains f u l l - i n f o r m a t i o n and does not y i e l d r e s u l t s d i f f e r e n t from Hansen and Sargent. Chow uses the parameterization of standard optimal c o n t r o l theory, where the model of the movement of input and output prices i s incorporated into the " t r a n s i t i o n equation", which represents the dynamic constraint f o r the fir m i n i t s optimization problem. When the parameters of the system are estimated, i t remains true that there are cross-parameter r e s t r i c t i o n s t e s t i n g the j o i n t hypothesis of the model of the f i r m and r a t i o n a l expectations. Full-i n f o r m a t i o n methods have so f a r only been applied to problems where the firm's objective function can be described i n l i n e a r or quadratic terms. This leads to l i n e a r input demand functions, at le a s t i n the reduced form parameters, although the cross-equation r e s t r i c t i o n s are non-linear and extremely complex 1 4 even f o r models with only two inputs. Linear-quadratic objective functions also allow the modeller to invoke the p r i n c i p l e of "certainty-equivalence"; the s o l u t i o n to the firm's optimal c o n t r o l problem i n the uncertain world i s the same as i t would be had the f i r m perfect f o r e s i g h t . Comparisons of the two methods as applied to dynamic models of input demand are made by West ( 1 9 8 6 ) and Prucha and N a d i r i ( 1 9 8 6 ) . West compares l i m i t e d - and f u l l - i n f o r m a t i o n methods numerically, and finds that the f u l l - i n f o r m a t i o n method lowers standard errors only s l i g h t l y , and that i t s parameter estimates tend to be more biased than limited-information estimates when the model i s misspecified. Monte Carlo comparisons of the two techniques by Prucha and N a d i r i , on the other hand, f i n d "considerable gains i n s t a t i s t i c a l e f f i c i e n c y " ( p . 2 0 9 ) from using f u l l - i n f o r m a t i o n methods. 1.4 Modelling Investment with Rational Expectations Ful l - i n f o r m a t i o n estimation of a dynamic model of input demand i s usually c a r r i e d out with the assumption of a linear-quadratic objective function for the firm.*' If y i s a scalar output and i s a column-vector of inputs then the usual form of the production function, excluding adjustment costs i s See Epstein and Yatchew ( 1 9 8 5 ) , Hansen and Sargent ( 1 9 8 0 , 1 9 8 1 ) , Meese ( 1 9 8 0 ) , Sargent ( 1 9 7 8 ) , and West ( 1 9 8 6 ) for examples. 15 ' y t - a ' X t + X t ' A x t / 2 ( 1 - 2 ) where a i s a vector and A i s a symmetric and negative d e f i n i t e matrix. Convex adjustment costs are also s p e c i f i e d as quadratic, and could be represented by ( x t " X t - l ) , B ( X t " V l ) / 2 ( 1 - 3 ) where B i s a symmetric matrix. The t h e o r e t i c a l r e s u l t s of such a model of the f i r m under r a t i o n a l expectations are described by Lucas and Prescott (1971), who are p r i m a r i l y concerned with the equilibrium p r i c e of c a p i t a l , and by Sargent (1979, Chapter 14, and 1981) who considers i n p a r t i c u l a r the optimal decision rule f or the firm. The f i r s t empirical use of t h i s model i s Sargent (1978) , who considers the demand f o r labour, both straight-time and over-time, when there are adjustment costs present. Sargent takes wages as being exogenous with respect to labour demand (this assumption i s j u s t i f i e d on the basis of c a u s a l i t y t e s t s with which Sargent begins the paper). Kennan (1988), on the other hand, estimates a model using Sargent's f i r m together with endogenous labour supply, where labour suppliers also have linear-quadratic objective 6 functions. Kennan's model i s drawn from Sargent (1979, Chapter 16). See N i c k e l l (1986) f o r a complete survey of dynamic models of labour Meese (1980) uses f u l l - i n f o r m a t i o n techniques to estimate the demand f o r c a p i t a l and labour by U.S. manufacturing (using quarterly data from 1947 to 1974) according to a linear-quadratic model. A f t e r j u s t i f y i n g the modelling of user costs and wages as exogenous with respect to input l e v e l s , he simultaneously estimates a four equation model: one equation each for demand f o r labour and demand f o r c a p i t a l , and a b i v a r i a t e autoregressive model of user costs and wages. The hypothesis of r a t i o n a l expectations imposes r e s t r i c t i o n s between parameters of the former two equations and the l a t t e r . The r e s t r i c t i o n s are highly non-linear. Meese remarks (pp.149-50): The estimation of the constrained version of the model...is a d i f f i c u l t task. Few software routines are capable of estimating a model of such complexity... Estimation i s c a r r i e d out by appending a "concentrated l i k e l i h o o d function" with a penalty function,. where the penalty function weights the various r e s t r i c t i o n s of the model. A l i k e l i h o o d r a t i o t e s t of the model, comparing the r e s t r i c t e d version to what amounts to an u n r e s t r i c t e d four equation vector autoregression, r e j e c t s the t h e o r e t i c a l r e s t r i c t i o n s at any s i g n i f i c a n c e l e v e l greater than 2%. Epstein and Yatchew (1985) take the t h e o r e t i c a l model used by Meese, and f i n d a reparameterization of the estimating equations that somewhat s i m p l i f i e s the estimation of the r e s t r i c t e d model. demand. It i s t h i s s i m p l i f i e d version that i s estimated, without amendment, with Canadian data i n t h i s t h e s i s i n Chapter 3, so a f u l l discussion of the Epstein and Yatchew method i s deferred u n t i l l a t e r . We now turn our attention to a l t e r n a t i v e methods of estimating dynamic models of input demand with r a t i o n a l expectations. A model of Tobin's (1969) has generated a method of modelling investment known as "q-theory". Tobin's q i s the r a t i o of the nominal market value of a f i r m to the nominal value of the firm's c a p i t a l stock evaluated at replacement cost. A value of q greater than one should lead to p o s i t i v e net investment, since the value of the new c a p i t a l w i l l be greater than i t s cost. The rate of investment, assuming there are convex adjustment costs, should then be p o s i t i v e l y r e l a t e d to the current value of q. It i s assumed that the value of the firm's equity captures the market's expectations about the future value of c a p i t a l . Hayashi (1982) makes two important observations. F i r s t , to model investment we should use "marginal q" rather than "average q" as the explanatory variable, where the former i s the marginal change i n the market value of the f i r m f or an a d d i t i o n a l unit of c a p i t a l divided by the p r i c e of a unit of c a p i t a l , and the l a t t e r i s the t o t a l market value of the f i r m divided by the t o t a l value of the c a p i t a l stock at replacement values. Second, a q-theory model using marginal q and the n e o c l a s s i c a l model with adjustment 18 7 costs and r a t i o n a l expectations are equivalent theories . Q-theory models of investment have been estimated by Hayashi, Abel (1980), Summers (1981), and McKibbin and S i e g l o f f (1988). A problem with empirical a p p l i c a t i o n of q-theory i s that: ...(with) the use of stock market valuation to i n f e r investor perception of p h y s i c a l investment opportunities...the information must be taken i n toto. But the information relevant to investment may be overshadowed by the v o l a t i l i t y of the "noise" i n stock market f l u c t u a t i o n s . (Bosworth commenting on Summers (1981), p. 130). A further problem, mentioned by Abel (1980, p. 77) i s that i f we r e l y on v a r i a t i o n s i n the stock market to explain f l u c t u a t i o n s i n investment, i t "begs the question, since i t does not explain what [factors determine values in] the stock market". A second a l t e r n a t i v e method to modelling dynamic input demand i s that r e f e r r e d to e a r l i e r as the "limited-information" method, where the Euler equations of the firm's optimal c o n t r o l problem are estimated d i r e c t l y . Although some useful information i s s a c r i f i c e d when t h i s technique i s used, there i s the advantage that we need not r e s t r i c t ourselves to technologies that are linear-quadratic. This allows the possible use of production functions that more c l o s e l y f i t the f a c t s . Hayashi c r e d i t s Lucas and Prescott (1971) for t h i s i n s i g h t , although they d i d not put i t i n these terms. Pindyck and Rotemberg (1983a, b) apply limited-information techniques to a model with a translog r e s t r i c t e d cost function, quadratic adjustment costs, and r a t i o n a l expectations. The model i s estimated with annual U.S. manufacturing data (1948-71 f o r (1983a) and 1949-76 for (1983b)). Shapiro (1986a) uses the same techniques, but with a Cobb-Douglas production function, f o r quarterly U.S. manufacturing data from 1955-80. Kokkelenberg and Bischoff (1986) use a polynomial approximation to a short-run va r i a b l e cost function on quarterly U.S. manufacturing data from 1959 to 1977. While there are no studies of t h i s type, of which we are aware, that apply to Canadian data, Carmichael, Mohnen, and Vigeant (1989) apply a translog variable cost function to Quebec manufacturing data (annual 1962-83). Their t e s t s f a i l to r e j e c t the r e s t r i c t i o n s imposed by the model. It i s i n t e r e s t i n g to note here t h e i r r e s u l t s f or the e l a s t i c i t y of c a p i t a l with respect to user costs, since t h i s s t a t i s t i c w i l l also be estimated i n t h i s t h e s i s . They f i n d the "impact e l a s t i c i t y " to be - 0.0 98 and the long-run e l a s t i c i t y (for a shock to user costs that i s permanent and immediately recognized as permanent) to be -0.271. The f i n a l approach to estimating dynamic input demands with r a t i o n a l expectations i s from Epstein and Denny (1983). A f l e x i b l e f u n c t i o n a l form i s chosen f o r the value function of a firm's va r i a b l e cost minimization problem, where adjustment costs are present. A l i m i t e d s p e c i f i c a t i o n of expectations i s allowed; i t i s supposed that r e a l input p r i c e s follow f i r s t - o r d e r autoregressive processes. While t h i s may not be consistent with " f u l l y r a t i o n a l " expectations, i n that there may be other information a v a i l a b l e useful f o r forecasting input p r i c e s , i t c l o s e l y approximates what i s u sually s p e c i f i e d i n p r a c t i c e i n r a t i o n a l expectations models anyway. A more complete discussion of t h i s model i s found i n Chapter 4 of t h i s t h e s i s . Neither the linear-quadratic r a t i o n a l expectations model of dynamic input demand, nor the model of Epstein and Denny, have 9 been estimated, to our knowledge, with Canadian aggregate data. One of the two p r i n c i p a l contributions of t h i s t hesis i s to estimate, and compare, these two methods of estimating input demands, using Canadian manufacturing data. 1.5 Tax Policy And Investment Since the introduction of n e o c l a s s i c a l theories of investment, which provided a l i n k between user costs of c a p i t a l , and therefore corporate tax p o l i c y , to investment, researchers have been using these models to consider the e f f e c t s of various tax p o l i c i e s on investment. H a l l and Jorgenson (1967) i s frequently c i t e d as the seminal Morrison and Berndt (1981) estimate a s t a t i c expectations version of a model s i m i l a r to Epstein and Denny's. 9 Bernstein (1986) estimates a s t a t i c expectations version of Epstein and Denny's model with the pooled data of some Canadian firms engaged i n research and development. a r t i c l e i n t h i s f i e l d . The number of studies on U.S. tax p o l i c y and investment since H a l l and Jorgenson i s immense; a survey of n e o c l a s s i c a l models of investment and tax p o l i c y i n the U.S. i s given by Chirinko (1986, 1987) . Chirinko and Eisner (1983) compare the empirical predictions of a number of U.S. macroeconomic models regarding tax p o l i c y and investment. Here we w i l l confine the discussion to Canadian studies of investment, which w i l l l a t e r be compared to the r e s u l t s obtained i n t h i s t h e s i s . B i r d (1980) c l a s s i f i e s the various studies that have been done i n Canada by three types: survey, econometric, and quasi-empirical. The two major surveys on tax p o l i c y and investment i n Canada are H e l l i w e l l (1966) and the Tax Measures Review Committee (1975). H e l l i w e l l considers the behaviour of 70 large firms, 35 of which are i n manufacturing, the others deal i n resources or services. These firms are taken from those which were interviewed by the Royal Commission on Banking and which were also sent questionnaires by the Royal Commission on Taxation. Two tax i n i t i a t i v e s are examined. One i s a 1961 change to depreciation allowances, which allowed depreciation at double the normal rates i n the year an asset was purchased, with normal rates of depreciation i n following years. This provision l a s t e d u n t i l January 1, 1964. Summarizing the r e s u l t s of the surveys, H e l l i w e l l (1968, p. 128) says the measure was not "...thought by firms to have had a noticeable influence on t h e i r investment expenditures". The other i n i t i a t i v e i s a 1963 proposal to allow 50% s t r a i g h t - l i n e depreciation on machinery and equipment for c a p i t a l purchased i n the two years commencing June 14, 1963 by firms which are e i t h e r 25% Canadian owned and c o n t r o l l e d or are involved i n manufacturing and processing i n designated areas of slow growth. The measure was to expire i n June 1965, although i t was l a t e r extended to the end of 1966. H e l l i w e l l (1966, pp. 170-72) provides a number of responses by managers which i n general suggest that the main impact of t h i s incentive was to change the timing of investment pro j e c t s . For example, one company spokesman sai d : Although we wouldn't undertake a project because of the accelerated depreciation, we probably w i l l order our equipment e a r l y to allow us to take whatever advantage i s obtainable. The Tax Measures Review Committee was e s p e c i a l l y i n t e r e s t e d i n the e f f e c t s on firms of the investment tax incentives introduced i n the 1972 fe d e r a l budget. These incentives consisted of accelerated depreciation allowances and lower corporate tax rates i n the manufacturing sector. With 1,288 firms responding to t h e i r survey, they found: ...83 per cent of the respondents a n t i c i p a t e d that the tax measures would have some p o s i t i v e impact on t h e i r operations. ...Increased investment expenditures as a r e s u l t of the tax measures were an t i c i p a t e d by 47 per 10 cent of the respondents... Tax Measures Review Committee (1975, p. 9) The p o s i t i v e responses obtained are s u r p r i s i n g given the survey r e s u l t s of H e l l i w e l l , and as we s h a l l see l a t e r are not consistent with empirical studies. May (1979, p. 73) claims "The findings of the Committee were greeted with a good deal of skepticism by both the p r o f e s s i o n a l and p o l i t i c a l communities". It i s well known that economists are often s k e p t i c a l regarding any r e s u l t s of surveys. By quasi-empirical studies of investment B i r d (1980, p. 42) has i n mind research that "...uses numbers, but i n a much less formal way than i n the econometric studies". Three examples are Hyndman (1974), Harman and Johnson (1978), and Johnson and Scarth (1979) . Hyndman's concern i s the e f f e c t s of the 1972 corporate tax changes for manufacturing firms that were also the focus of the Tax Measures Review Committee study discussed e a r l i e r . Hyndman does not a c t u a l l y c a l c u l a t e the e f f e c t s on investment, but rather assesses the e f f e c t s of the tax changes on the costs of production, leaving the reader to i n f e r what the o v e r a l l e f f e c t s might be. He says the 1972 changes lowered user costs of machinery and equipment i n manufacturing by at most 20%, which he claims increases the p r i c e of f i n a l output r e l a t i v e to costs by about 3%. Harman and Johnson estimate the e l a s t i c i t y of investment with respect to user costs of c a p i t a l using a model f i r s t suggested by Coen (1971), i n which investment depends on new orders, cash flow, past investment, and the user cost r e l a t i v e to wages. The r e s u l t s are used to c a l c u l a t e the impacts of various investment incentives from the 1963 fe d e r a l budget to the 1972 budget. Since the only figures reported are the present value of induced investment from the budget i n i t i a t i v e s i t i s d i f f i c u l t to i n f e r the relevant e l a s t i c i t i e s . Johnson and Scarth, l i k e Harman and Johnson, place much emphasis on the r a t i o of induced investment to the l e v e l of the "tax expenditure" by the government. H a l l and Jorgenson's (1967) model of investment i s used to c a l c u l a t e the e f f e c t s of an investment tax c r e d i t and a lowering of the corporate tax rate. The model of investment i t s e l f i s not a c t u a l l y estimated, but rather parameter values are imposed, so the impact of investment incentives on investment i s given a p r i o r i . Turning to the f i n a l category of research on investment behaviour, econometric studies on Canada are Wilson (1967), McFetridge and May (1976), and Braithwaite (1983). Wilson's econometric model of investment assumes putty-clay c a p i t a l 1 1 and a d i s t r i b u t e d lag l i n k i n g completed investment projects to c a p i t a l appropriations. At any time the optimal c a p i t a l stock depends on user costs of c a p i t a l , a v a i l a b i l i t y of corporate funds, and output. A range of reduced forms i n v o l v i n g the aforementioned variables are estimated. The long run e l a s t i c i t y of investment to the i n t e r e s t rate i s estimated as -0.67. Similar models estimated with U.S. data found somewhat C a p i t a l i s putty-clay i n the p a r t i c u l a r sense that f a c t o r proportions are f i x e d f o r completed projects, but not f o r "backlogged" projects (Wilson (1967, p. 36)). lower e l a s t i c i t i e s : Jorgenson (1963) obtained -0.38 and Bischoff (1971) found -0.23. Wilson also found, as d i d the U.S. studies, that the peak investment response to an i n t e r e s t rate shock was one year following the shock (the model i s q u a r t e r l y ) . McFetridge and May estimate a model of investment s i m i l a r to Jorgenson's, with investment being l o g - l i n e a r i n output and r e l a t i v e input p r i c e s and depending on an ad hoc lag structure. The impact e l a s t i c i t y of c a p i t a l stock with respect to the user cost of c a p i t a l , a^ i n t h e i r notation, i s -0.08, and the long run 12 e l a s t i c i t y , Tf^, i s -0.43. The r e s u l t s are used to analyse the accelerated depreciation changes of 1972. Their estimate of the extra gross investment induced by the change i n tax p o l i c y i s only about one-quarter the estimate of the Tax Measures Review 13 Committee r e f e r r e d to e a r l i e r . Braithwaite embeds his model of investment i n the Economic Council of Canada's macroeconomic model CANDIDE 2.0. Investment i s modelled as depending on d i s t r i b u t e d lags of the value of output r e l a t i v e to the user cost of c a p i t a l , and l e v e l s of the c a p i t a l stock; i . e . no production function i s e x p l i c i t l y described. The In Chapters 3 and 4 of t h i s t h e s i s the e l a s t i c i t y of c a p i t a l with respect to user cost i s estimated and can be compared d i r e c t l y to McFetridge and May's r e s u l t s . 13 May (1979) remarks that the Tax Measures Review Committee claimed larger e f f e c t s of tax p o l i c y on investment than any empirical study. 26 estimates of his investment equation are d i f f i c u l t to i n t e r p r e t . On the other hand, a simulation of Braithwaite's which w i l l be of i n t e r e s t to us i s an increase i n the investment tax c r e d i t . He considers a permanent increase, i n 1980, by a factor of 1.8 i n the investment tax c r e d i t . Since i n 1980 the base rate of the investment tax c r e d i t i n Canada was 7%, the experiment involves increasing the base rate to 12.6%. The e f f e c t on investment i n machinery and equipment i n manufacturing, r e l a t i v e to the base 14 case, expressed i n terms of m i l l i o n s of 1971 d o l l a r s i s : year 1980 1981 1982 1983 1984 1985 change +30 +82 +126 +134 +125 +68 In 1983, where the e f f e c t on investment peaks, the e f f e c t i s 3.2% of the base case gross investment. Each of the studies described i n t h i s section take quite d i f f e r e n t approaches to estimating the e f f e c t s of tax p o l i c y on investment. B i r d (1980, p. 46) remarks "anyone t r y i n g to discern the e f f e c t s of incentives on investment from the studies reviewed above must f e e l as though he has wandered into the Tower of Babel". Yet with the exception of the Tax Measures Review Committee there does seem to be some consensus that the demand f o r c a p i t a l i s f a i r l y i n e l a s t i c with respect to the user cost. F e l d s t e i n (1982) eloquently states the case f o r considering a 1 4 B r a i t h w a i t e (1983, p. 67) wide range of t h e o r e t i c a l models of investment when attempting to ask any question, and for looking for r e s u l t s which seem to be invariant to model s e l e c t i o n . In t h i s t hesis two models of input demand w i l l be estimated. Each of the models assumes firms face adjustment costs when changing input l e v e l s . For each of the models we obtain estimates under a l t e r n a t i v e a p r i o r i s p e c i f i c a t i o n s of s t a t i c and non-static expectations. It i s c l e a r that the models used here w i l l not exhaust a l l possible ways of c a l c u l a t i n g the e f f e c t s of the r e n t a l rate on investment, but i t i s hoped that the r e s u l t s w i l l complement the other studies described above. We w i l l be i n t e r e s t e d i n whether r a t i o n a l expectations models of input demand with adjustment costs generate e l a s t i c i t i e s very d i f f e r e n t from those found by others. The case can also be made that the models used here represent something of an improvement over previous studies; techniques designed to improve the way input demand models are estimated, which have only very recently been developed, are put to use. In Chapter 2 , the data to be used i n estimation i s described i n d e t a i l . The linear-quadratic model i s described i n d e t a i l and estimated i n Chapter 3. Some analysis of the time-series properties of the data i s undertaken to ensure that the data are compatible with the t h e o r e t i c a l model to which they w i l l be applied. The s i m p l i f i e d estimation procedure of Epstein and Yatchew (1985) i s used. The method of doing p o l i c y analysis with the model i s described, and some simulations are c a r r i e d out which 28 consider the e f f e c t s of changes i n user costs on investment. In Chapter 4 the model of Epstein and Denny (1983) i s described and estimated. Simulations s i m i l a r to those i n Chapter 3 are done for the purposes of comparison, and provide an i n d i c a t i o n of the dependence of p o l i c y simulation r e s u l t s on the p a r t i c u l a r model of dynamic input demand chosen. The r e s u l t s of these three chapters are useful f or three reasons. F i r s t , they provide us with more information on the usefulness of such models and whether there are d i s t i n c t advantages i n using one type of s p e c i f i c a t i o n rather than another. Chapter 4 contains an a p p l i c a t i o n of Davidson and MacKinnon's (1981) "P t e s t " , which, f o r each of the two models, evaluates the model where the other model i s taken as the a l t e r n a t i v e hypothesis.Second, they w i l l provide some estimates of the e f f e c t s of the r e n t a l rate of c a p i t a l on investment, which can then be compared to previous studies. Third, we w i l l obtain some empirical evidence on the dependence of the reduced form parameters of input demand equations on expectations. Alan Blinder has remarked: 1^ The Lucas c r i t i q u e may be correct, but I have seen no persuasive evidence i n any sphere to ind i c a t e that i t i s e m p i r i c a l l y important. The empirical case i s yet to be made. The b i g question i s whether changes i n p o l i c y regimes cause large changes i n c o e f f i c i e n t s . Maybe they cause just very t i n y changes. In Klamer (1983, p. 166). We w i l l provide evidence on the question of the degree to which changes i n the 'tax p o l i c y regime' a f f e c t the reduced form c o e f f i c i e n t s of input demand equations. Chapter 5 concludes the t h e s i s . The r e s u l t s , and the comparisons with previous studies, w i l l be summarized, and possible future research w i l l be described. 30 CHAPTER 2 The Data In t h i s chapter a l l data that w i l l be used i n estimating the input demand models i n Chapters 3 and 4 are described. 2.1 Wages and Rental Rates Table 2.1 l i s t s the data used i n t h i s t h e s i s . The nominal r e n t a l rate f o r c a p i t a l i s from the Economic Council of Canada CANDIDE 2.0 Database, and i s described i n d e t a i l by Braithwaite (1983). The following d e r i v a t i o n of the i m p l i c i t r e n t a l rate of c a p i t a l i s taken from Boadway (1980). Imagine a p e r f e c t l y competitive f i r m which uses c a p i t a l , k, to produce output, y. C a l l the marginal product of c a p i t a l MPK and the purchase p r i c e of a unit of c a p i t a l q. According to the n e o - c l a s s i c a l theory of investment i n the absence of any adjustment lags or adjustment costs i n c a p i t a l stock, the f i r m w i l l purchase units of c a p i t a l up to the point where the p r i c e of a unit of c a p i t a l i s equal to i t s net-of-tax present discounted value of marginal revenue product: 00 q f c = S P s M P K s ( l - U s ) e " ( R + 6 ) ( S _ t ) d s + q t u t Z t ( l - I T C t ) + q t I T C t (2.1) where p i s the p r i c e of output at time s, u i s the p r o f i t s tax s rate, R i s the i n t e r e s t rate, 6 i s the depreciation rate on c a p i t a l , Z i s the present discounted value of deductions allowed for depreciation and i n t e r e s t costs, and ITC i s the investment tax c r e d i t rate. Note that i t i s presumed here, as i s a c t u a l l y the case i n Canada, that the amount of c a p i t a l e l i g i b l e f o r depreciation and i n t e r e s t allowances i s reduced by the amount of the investment tax c r e d i t . D i f f e r e n t i a t i n g equation (2.1) with respect to t gives: q =(R+5)q (1-u Z (1-ITC J ) - I T C - (1-uJ p MPK +q u Z (1-ITC )+o_ITC t t t t t t t t t t t t t t t where q f c i s the time d e r i v a t i v e of q f c. Solving f or MPK^ gives: MPK = ( (R + 5) q - q ) (1 - u Z ) (1 - ITC ) / ( (1 - u. ) p̂ .) t t t t t t t t (2.2) The r i g h t hand side of t h i s equation i s the r e a l i m p l i c i t r e n t a l rate of c a p i t a l , which i s the net-of-tax cost of using a unit of c a p i t a l f o r one time period. At each point i n time the desired c a p i t a l stock i s the l e v e l where the marginal product of c a p i t a l equals the r e a l i m p l i c i t r e n t a l rate, hereafter denoted r . In the seri e s f or the nominal r e n t a l rate i n Canadian manufacturing given i n Table 2.1, the investment goods p r i c e index q^ i s set equal to 1 i n 1971. The Economic Council of Canada provides data f o r the nominal r e n t a l rate on c a p i t a l ; t h i s s e r i e s w i l l be d e f l a t e d by the manufacturing sector output p r i c e index p̂ _, also l i s t e d i n Table 2.1, to generate the r e a l s e r i e s r̂ _ f o r use i n the model of Chapter 3, and the seri e s w i l l be d e f l a t e d by the manufacturing sector material input p r i c e index m̂  when used i n the model of Chapter 4. Since p i s normalized to equal 1 i n 32 1971, r i s t h e n t h e a f t e r - t a x a n n u a l c o s t i n 1971 m a n u f a c t u r i n g o u t p u t d o l l a r s o f u s i n g one 1971 d o l l a r ' s w o r t h o f m a c h i n e r y and equipment. B r a i t h e w a i t e (1983, pp 9-15) d e s c r i b e s t h e d a t a i n g r e a t d e t a i l , so o n l y a b r i e f d e s c r i p t i o n i s g i v e n h e r e . We n o t e a t t h e o u t s e t t h a t t h e r e n t a l r a t e d a t a can o n l y a p p r o x i m a t e t h e t r u e r e n t a l r a t e , due t o t h e many c o m p l e x i t i e s o f t h e t a x system, and due t o o u r i g n o r i n g some o f t h e p o s s i b l e e f f e c t s on f i r m b e h a v i o u r o f t h e c o r p o r a t e t a x , w h i l e t a k i n g f u l l a c c o u n t o f o t h e r s . The i n t e r e s t r a t e R i s a w e i g h t e d average o f t h e e x p e c t e d e q u i t y c o s t o f c a p i t a l and t h e a f t e r - t a x bond r a t e , t h e w e i g h t s b e i n g d e t e r m i n e d by t h e h i s t o r i c a l e q u i t y s h a r e o f t o t a l c a p i t a l . 1 The c o r p o r a t e t a x r a t e u i s t h e " e f f e c t i v e " t a x r a t e , o b t a i n e d by d i v i d i n g t o t a l t a x e s p a i d by n e t t a x a b l e income. I d e a l l y , i n t r y i n g t o e v a l u a t e t h e e f f e c t s o f t h e r e a l r e n t a l r a t e on i n v e s t m e n t , we would want t o use a measure o f t h e marginal e f f e c t i v e t a x r a t e r a t h e r t h a n t h e ave r a g e r a t e . T h i s d a t a was u n a v a i l a b l e t o u s ; f o r some r e c e n t r e s e a r c h on how e f f e c t i v e m a r g i n a l t a x r a t e s might be c a l c u l a t e d see Boadway (1987) o r Boadway, B r u c e , and M i n t z (1987). There e x i s t s a body o f r e s e a r c h on t h e e f f e c t s o f changes i n t h e c o r p o r a t e t a x s y s t e m on t h e f i n a n c i n g d e c i s i o n s o f f i r m s , b u t i t i s beyond t h e scope o f t h i s t h e s i s t o a t t e m p t t o i n c o r p o r a t e any such e f f e c t s i n o u r d a t a . See Auerbach (1983) f o r a s u r v e y o f t h i s r e s e a r c h . 33 Equation (2.2) describing the r e n t a l rate of c a p i t a l contains the assumption that there i s f u l l l o s s - o f f s e t t i n g . In f a c t i n Canada l o s s - o f f s e t t i n g i s imperfect. Mintz (1988) provides some estimates of how t h i s imperfection might a f f e c t e f f e c t i v e tax rates i n Canada. The investment tax c r e d i t (ITC) was introduced i n the federal government budget of 1975. The terms of the ITC are described i n d e t a i l by Timbrell (1975). Boadway and Kitchen (1984, p.146) describe the changes to the ITC up to 1984. B r i e f l y , the ITC i s a tax c r e d i t on gross investment i n structures and machinery and equipment (note that i n t h i s t h e s i s c a p i t a l w i l l r e f e r simply to machinery and equipment). The measure of the value of c a p i t a l stock used f o r depreciation allowances i s reduced by the amount of the investment tax c r e d i t . When introduced, the base rate of the ITC was f i v e per cent. At the end of 1978 the base rate was increased to seven per cent, and i t remained at t h i s l e v e l through 1984. In equation (2.1) the e f f e c t i v e r e a l r e n t a l rate on c a p i t a l i s reduced by exactly the ITC rate; i f r i s the r e a l r e n t a l rate before the ITC i s introduced, i t i s - ITC) a f t e r the ITC i s introduced. This i s the r e s u l t of assuming that the purchase p r i c e of c a p i t a l goods, q, i s determined i n a competitive i n t e r n a t i o n a l market, of which the Canadian manufacturing sector i s but a small part. If we imagine that c a p i t a l i s a c t u a l l y rented or leased, then we could describe the model as assuming that the lessee bears the burden of the corporate tax and receives the benefits of the investment tax c r e d i t . This has been the working assumption i n other studies of taxation and investment; see, for example, the general equilibrium analysis i n Hamilton and Whalley (1989, esp. pp.383-4). Note that we ignore the treatment by foreign governments of corporate income earned i n Canada by multi-national firms. This s i m p l i f i e s the construction of the r e a l r e n t a l rate s e r i e s , but t h i s s i m p l i f i c a t i o n i s perhaps j u s t i f i a b l e ; see Hartman (1985) f or an explanation of why the foreign tax rates can be i r r e l e v a n t to the investment d e c i s i o n of a multi-national firm. The nominal wage data are also from the CANDIDE 2.0 Database, and are average hourly earnings i n manufacturing. In Chapter 3 t h i s s e r i e s i s de f l a t e d by the manufacturing output p r i c e index p̂ _, so that ŵ_ i s average hourly earnings i n 1971 manufacturing output d o l l a r s . The output p r i c e index p^ and the material input p r i c e index m are constructed from S t a t i s t i c s Canada's Input-Output data (annual catalogues 15-201E and 15-202E), prices being implied and revealed by d i v i d i n g the current d o l l a r s t a t i s t i c s with the constant d o l l a r s t a t i s t i c s . Prices f o r each sector are weighted i n the index by the sector's share of t o t a l manufacturing output i n that year. 2.2 C a p i t a l and Labour Inputs C a p i t a l and labour inputs f o r manufacturing are l i s t e d i n Table 2.1. C a p i t a l , k̂ _, i s machinery and equipment i n Canadian manufacturing as given by the Economic Council of Canada, and i s 35 measured i n terms o f m i l l i o n s o f 1971 d o l l a r s ' w o r t h . B r a i t h e w a i t e (1983, pp. 15-16) d e s c r i b e s how i t i s c a l c u l a t e d . Labour, 1^, i s measured i n terms o f m i l l i o n s o f manhours, and i s a l s o t a k e n f r o m t h e Economic C o u n c i l o f Canada's d a t a b a s e . Note t h a t t h e d a t a a r e c o n s t r u c t e d s u c h t h a t r t ^ t ' a n n u a l e x p e n d i t u r e s on c a p i t a l s e r v i c e s , and w t l t ' a n n u a l e x p e n d i t u r e s on l a b o u r s e r v i c e s , a r e b o t h measured i n terms o f m i l l i o n s o f 1971 d o l l a r s (note t h a t t h e d e f l a t o r i s a m a n u f a c t u r i n g s e c t o r o u t p u t p r i c e i n d e x ) . 2.3 Output The r e a l o u t p u t o f t h e m a n u f a c t u r i n g s e c t o r , y , i s f r o m S t a t i s t i c s Canada's Input-Output S t a t i s t i c s of the Canadian Economy, v a r i o u s i s s u e s f r o m 1961 t o 1984. S i n c e k f c and 1 a r e c a p i t a l and l a b o u r i n t h e " m a n u f a c t u r i n g s e c t o r " , o u t p u t i s t a k e n f r o m t h e t o t a l o u t p u t o f t h e m a n u f a c t u r i n g s e c t o r , and does n o t s i m p l y r e p r e s e n t m a n u f a c t u r i n g goods. I n s p e c t i o n o f t h e I n p u t - O u t p u t t a b l e s r e v e a l s t h a t t h e m a n u f a c t u r i n g s e c t o r p r o d u c e s some non-manufactured goods (e.g. some s e r v i c e s ) and t h a t some o t h e r s e c t o r s produce m a n u f a c t u r e d goods (e.g. t h e a g r i c u l t u r e and f o r e s t r y s e c t o r s produce some m a n u f a c t u r e d g o o d s ) . Here we make t h e d e f i n i t i o n o f o u t p u t c o n s i s t e n t w i t h t h e d e f i n i t i o n o f i n p u t s . Note a l s o t h a t t h e i m p l i c i t o u t p u t p r i c e i n d e x i s d e r i v e d f r o m t h e o u t p u t o f t h e m a n u f a c t u r i n g s e c t o r , and n o t f r o m t h e o u t p u t o f m a n u f a c t u r e d goods p r o d u c e d by t h e e n t i r e economy. 36 TABLE 2.1 Data Year P y N w 1 N r k m 1961 0. 8315 29649 .6 2 .18 2789 .113 9049. 3 0 .7978 1962 0. 8393 32324 .9 2 .24 2903 .121 9300. 3 0 .8130 1963 0. 8486 34734 .9 2 .30 3010 .121 9531. 0 0 .8254 1964 0. 8588 37951 .9 2 .36 3201 .114 10207 . 5 0 .8400 1965 0. 8714 41292 .6 2 .63 3184 .119 11055. 8 0 .8596 1966 0. 8926 44149 .2 2 .78 3377 .122 12183. 2 0 .8866 1967 0. 9104 45235 .4 2 .99 3367 .132 12954. 7 0 . 9067 1968 0. 9254 48126 .1 3 .23 3328 .138 13327. 5 0 .9240 1969 0. 9524 51338 .9 3 .41 3445 .144 13907. 3 0 . 9574 1970 0. 9792 50617 .6 3 .65 3370 .149 14742. 6 0 .9834 1971 1. 0000 53479 .1 3 .91 3346 .143 15336. 4 1 .0000 1972 1. 0417 57571 .9 4 .18 3466 .158 15811. 3 1 .0478 1973 1. 1449 62669 .9 4 .57 3630 .135 16696. 2 1 .1694 1974 1. 3677 65105 .2 5 .27 3702 .148 17813. 4 1 .3894 1975 1. 5331 61241 .8 6 .13 3513 .163 18715. 3 1 .5391 1976 1. 6209 65056 .3 6 .92 3599 .164 19404. 4 1 .6136 1977 1. 7471 66613 .1 7 .62 3564 .190 20004. 6 1 .7349 1978 1. 9204 70069 .5 8 .10 3702 .211 20376. 6 1 . 9069 1979 2 . 1896 73236 .8 8 .81 3877 .236 21013. 3 2 .1880 1980 2 . 4646 71942 .4 9 .55 3920 .274 21982. 7 2 .4243 1981 2 . 7481 72967 .5 11 .01 3911 .344 23286. 0 2 .6130 1982 2 . 9396 65851 .7 12 .31 3517 .356 23805. 9 2 .7331 1983 3. 0411 69292 .3 13 .21 3484 .362 23698. 8 2 .7886 1984 3. 1385 76700 .9 13 .49 3621 .424 23603. 4 2 .9028 p i s a p r i c e index of goods and services produced by the Canadian manufacturing sector, y i s the quantity of such goods and services N measured i n m i l l i o n s of 1971 d o l l a r s , w i s average hourly earnings i n manufacturing measured i n current d o l l a r s , 1 i s N manhours of labour i n manufacturing measured i n m i l l i o n s , r i s the i m p l i c i t r e n t a l rate of machinery and equipment i n manufacturing measured i n the current d o l l a r cost of renting one 1971 d o l l a r ' s worth of c a p i t a l , k i s the stock of machinery and equipment i n manufacturing measured i n m i l l i o n s of 1971 d o l l a r ' s worth, and m i s a p r i c e index of material inputs used by the Canadian manufacturing sector. The source f o r p, m, and y i s S t a t i s t i c s Canada, Input-Output S t a t i s t i c s of the Canadian Economy (various issues), and the source f o r a l l other data i s the C AND IDE 2.0 databank of the Economic Council of Canada. 37 CHAPTER 3 The Linear Quadratic Model In t h i s chapter a model of input demand i s s p e c i f i e d and i s estimated with the data from the Canadian manufacturing sector described i n Chapter 2. Before the model i s described, i t i s necessary to examine some of the time seri e s properties of the data, since the model requires that the data s a t i s f y c e r t a i n conditions. 3.1 A Time Series Analysis of the Data We now examine some of the time seri e s properties of the wage and r e n t a l rate data. We ask i n turn (i) whether the two time serie s are stationary, and ( i i ) whether the wage and re n t a l rate are exogenous with respect to the l e v e l s of input demands. The motivation f o r t h i s examination i s that when the model of input demand i s l a t e r estimated we w i l l need some way to represent how firms might have formed expectations of future input p r i c e s . One possible method i s to assume that a l i n e a r time seri e s model of input p r i c e s can represent the way firms made expectations. We d e f l a t e the nominal r e n t a l rate and nominal wage by the manufacturing output p r i c e index given i n Table 2.1. Assume f o r now (the assumption i s j u s t i f i e d below) that the re n t a l rate and wage each follow a f i r s t - o r d e r autoregressive process, which we write as + e It (3.1) ' = v + 6 w t 2 22 t-1 + e 2t (3.2) 38 where i = I r 2 are random e r r o r s . Hansen and Sargent (1981, p.136) e s t a b l i s h the f a c t that a necessary condition for convergence of the firm's input decision rule i n the r a t i o n a l expectations l i n e a r quadratic model to be 5 5 estimated i n t h i s chapter i s {9^1 < (1+R)' and | &221 < (1+R) ' where R i s the discount rate. In t h i s section more stringent r e s t r i c t i o n s are tested: < 1 and | e 2 2 • < 1 " I n o t n e r words, i t w i l l be determined whether r and w can be considered stationary processes. From OLS estimates of (3.1) and (3.2) we obtain 6 = .8271 (.1222) and 9 ^ = .9211 (.0473) where standard errors are i n parentheses. F u l l e r (1976) demonstrates that the estimates of standard errors of 8 and 922 ^° n 0 t h a v e standard d i s t r i b u t i o n s under the n u l l hypotheses 9^= 1 and 922 = l f S O t h e e s t : " - m a t e s above cannot be used to e s t a b l i s h whether 6 and 8^2 are " s i g n i f i c a n t l y d i f f e r e n t " from 1. An appropriate t e s t for 8 < 1 and 922 < 1 °- e s c r :'- D e a^ by Dickey and F u l l e r (1979). Nelson and Plosser (1982) use t h i s t e s t to examine whether U.S. aggregate output i s stationary around a trend. The following equations are estimated by ordinary least squares: ( r t - Et-1> = "lO + a i i r t - l + a ! 2 ( r t - l " W + U l t ( 3 ' 3 ) ( W t " Wt-1> = a20 + a 2 1 W t - l + a 2 2 ( W t - l - W t - 2 } + U 2 t ( 3 - 4 ) Enough lagged values of the dependent variables of (3.3) and (3.4) are included on the r i g h t hand sides of these equations u n t i l the error terms are white-noise; one l a g turned out to be s u f f i c i e n t f o r these s e r i e s . OLS estimates of (3.3) and (3.4) are: r - r f c _ 1 = .024(.017) - .191(.131)r + .044(.238)<r - r ) ŵ  - ŵ  = .376 (.183) - .090 (.049)w^ , + . 381 (. 197) (ŵ  - w „) t t-1 t-1 t-1 X.—2. where standard errors are i n parentheses. The te s t of s t a t i o n a r i t y asks whether estimates of and &2\ a r e n e 9 a t i v e and s i g n i f i c a n t l y d i f f e r e n t from zero; i f so, the n u l l hypothesis of non-stationarity i s rejected. The te s t s t a t i s t i c i s (n - p)a^ (1 - o ^ ) 1 where i = 1,2 for r and w respectively, n i s the number of observations (22) and p i s the number of r i g h t hand side variables i n the regression (3). So f o r r t h i s t e s t s t a t i s t i c equals -3.788 and for w i t equals -2.751. F u l l e r (1976, p. 371) provides the d i s t r i b u t i o n of t h i s s t a t i s t i c (in h i s notation the t e s t s t a t i s t i c i s d i s t r i b u t e d as n(p - 1)). From h i s Table 8.5.1 we f i n d that the n u l l hypothesis of non-stationarity i s not rejected: the 0.1 s i g n i f i c a n c e l e v e l i s approximately -10.2.^ West (1988) i n his discussion of "near random-walk behavior" points out that with only a small number of observations, as we have here, i t w i l l be u n l i k e l y that non-stationarity can be rejected as a p o s s i b i l i t y even i f the true value of the lagged The only study of which I am aware that examines the s t a t i o n a r i t y of r e a l wages i n Canada i s the preliminary r e s u l t s given by Sigurdson and Stewart (1990) ; they suggest that r e a l wages i n Canada follow a random walk. A l t o n j i and Ashenfelter (1980) are unable to r e j e c t the hypothesis that the r e a l average hourly wage for the e n t i r e U.S. economy follows a random walk. Sargent (1978), Meese (1980), and Epstein and Yatchew (1985) are a l l empirical models of dynamic factor demand estimated with U.S. manufacturing sector data; none of these studies contains an e x p l i c i t t e s t of s t a t i o n a r i t y f o r r e a l input p r i c e s , although s t a t i o n a r i t y i s imposed by detrending the data. We now consider the properties of the r e n t a l rate series i n more d e t a i l . The r e s u l t of OLS estimation i s r = .0222 + .8271r t_ 1 (.0159) (.1222) with standard errors i n parentheses. The Durbin h s t a t i s t i c i s .1609, suggesting no s e r i a l c o r r e l a t i o n of the er r o r terms. Applying Gujarati's (1978, p. 246) "runs t e s t " , we f i n d 12 p o s i t i v e and 11 negative residuals (for the annual sample 1962 to 1984) and 11 runs. The 5% c r i t i c a l values for p o s i t i v e and negative s e r i a l c o r r e l a t i o n are 7 runs and 18 runs, respectively, so t h i s t e s t provides some further evidence for no s e r i a l term parameter i s as low as 0.8. c o r r e l a t i o n . The software SHAZAM (White 1978) provides the researcher with 7 d i f f e r e n t heteroskedasticity t e s t s t a t i s t i c s , a l l i n v o l v i n g an examination of the r e l a t i o n s h i p between estimated residuals (or some transformation of them) and the independent variables, or the predicted values of the regression (or some transformation of 2 them). A l l 7 t e s t s t a t i s t i c s are x with 1 degree of freedom. The maximum s t a t i s t i c of the set of seven f o r the r e n t a l rate equation i s 1.740, so one i s l e d to assume homoskedasticity. We now consider t e s t s of s t r u c t u r a l breaks i n the s e r i e s . Harvey (1981, pp. 151-4) discusses how one might analyse the cumulative sum and cumulative sum of squares of recursive r e s i d u a l s . Harvey's t - t e s t on recursive residuals (see h i s equation (2.10), p. 156) helps i d e n t i f y whether the recursive residuals tend to be the same sign. The t - s t a t i s t i c f o r our forward recursive estimation i s -0.6084 and for our backward recursive estimation i s +0.4285, each with 20 degrees of freedom. The p l o t s of the cumulative sum of squares also present no evidence of m i s s p e c i f i c a t i o n . F i n a l l y we consider the sets of sequential Chow tes t s and Goldfeld and Quandt t e s t s f o r s t r u c t u r a l break. with 23 observations the Chow t e s t s have 2 degrees of freedom i n the numerator and 19 i n the denominator. The 10% c r i t i c a l value for the F - s t a t i s t i c i s 2.61. This i s exceeded at one point i n the 2 sample, s p e c i f i c a l l y between 1972 and 1973. Examining the p l o t of Since we are t e s t i n g f o r s t r u c t u r a l break without asking a p r i o r i 42 the residuals of the AR(1) regression we f i n d the largest r e s i d u a l i n terms of absolute value occurs i n 1973, the f i r s t f u l l year accelerated depreciation allowances were i n e f f e c t (see Boadway and Kitchen (1984 pp. 128-9) f o r d e t a i l s ) . The Goldfeld and Quandt (1973) t e s t f o r s t r u c t u r a l break, which i s based on the r a t i o of the sum of squared residuals from regressions using the sample before a break and a f t e r , does not y i e l d a te s t s t a t i s t i c ( l i k e the Chow t e s t , an F - s t a t i s t i c ) at any possible break point that exceeds the 10% c r i t i c a l value. Of the Chow t e s t s , the Goldfeld and Quandt t e s t s , and various t e s t s on the pattern of recursive residuals, only the Chow t e s t gives any evidence of s t r u c t u r a l break. Casual observation of the data and of the residuals of the AR(1) estimation does not f i n d any obvious break i n the time s e r i e s . There are two implications of t h i s . F i r s t , we w i l l be able to use our en t i r e 1961 to 1984 data set when estimating the linear-quadratic r a t i o n a l expectations model, which requires that the input p r i c e s follow stationary processes (we examine the wage l a t e r ) . We need not estimate f o r d i f f e r e n t "regimes". Second, i t suggests that perhaps one should be wary of in t e r p r e t i n g any change i n corporate tax rules - say the introduction of accelerated depreciation allowances, or a change i n the rate of the investment tax c r e d i t , or a change i n the p r o f i t s tax rate - as a change i n the " p o l i c y regime". Sims (1982, where i t might occur, i t i s not s u r p r i s i n g that at le a s t one Chow test i s s i g n i f i c a n t at the 10% l e v e l . 43 p.108) writes: . . . i t i s a mistake to think that decisions about p o l i c y can only be described, or even often be described, as choice among permanent rules of behavior f o r the p o l i c y a u t h o r i t i e s . A p o l i c y action i s better portrayed as implementation of a f i x e d or slowly changing r u l e . This i s a possible way to think about corporate tax p o l i c y i n Canada. Suppose that the rule the government i s following i s to s t a b i l i z e to some degree the r e a l r e n t a l rate of c a p i t a l . I f , because of the design of the corporate income tax, high i n f l a t i o n causes the r e a l r e n t a l rate to r i s e beyond l e v e l s which the government thinks appropriate, s p e c i a l tax c r e d i t s and allowances may be introduced to o f f s e t the harmful e f f e c t s of i n f l a t i o n . Seen from t h i s angle, the introduction of the ITC might not represent a "regime change" at a l l , but rather i s simply a manifestation of a rule that was already i n place. This generates problems f o r those who wish to examine the e f f e c t s of one aspect of tax p o l i c y , f o r example the investment tax c r e d i t , i n p a r t i c u l a r . If one i s using a model where firms' expectations are presumed r a t i o n a l , how can one specify expectations f o r the counter-factual p o l i c y of no investment tax cre d i t ? Indeed, how does one speci f y the counter-factual p o l i c y i n the model? Should one assume that the other parts of the tax system remain unchanged, then one i s , as a counter-factual, considering what would have been a change i n regime. This problem i s examined further below, where estimated models are used i n simulations f o r some counter-factual time serie s of the re n t a l rate. We now examine the s t a t i o n a r i t y properties of the r e a l wage rate. The r e s u l t of OLS estimation i s w = .3613 + .9211wt 1 (.1750) (.0473) with standard errors i n parentheses. The Durbin h s t a t i s t i c i s 1.8012, suggesting there might be s e r i a l c o r r e l a t i o n i n the res i d u a l s . There are 13 p o s i t i v e , and 10 negative, residuals, with 9 runs. The 5% c r i t i c a l values f o r s e r i a l c o r r e l a t i o n are 7 and 18 (see Gujarati (1978, pp. 440-1)), so there i s some evidence against s e r i a l c o r r e l a t i o n as well. Casual observation of a plo t of the residuals y i e l d s no c l e a r evidence f o r or against s e r i a l c o r r e l a t i o n . 2 The values of the 7 % s t a t i s t i c s f o r heteroskedasticity given by SHAZAM range from 0.280 to 2.437. The 10% c r i t i c a l value with one degree of freedom i s 2.70 6, so there i s no strong evidence of heteroskedasticity. Regarding the recursive residuals, the cumulative sum of squares y i e l d s no casual evidence of m i s s p e c i f i c a t i o n . Harvey's t - t e s t of the cumulative residuals (1981, p. 156) y i e l d s a s t a t i s t i c of -0.953 f o r the forward recursive residuals, which does not lead one to suspect m i s s p e c i f i c a t i o n , but a s t a t i s t i c of -2.967 for the backward recursive r e s i d u a l s . This does suggest some sort of m i s s p e c i f i c a t i o n , but the plo t of residuals gives no cl e a r i n d i c a t i o n where any s t r u c t u r a l change i n the series might have taken place. The Chow t e s t s t a t i s t i c has an F d i s t r i b u t i o n with (2, 19) degrees of freedom. The highest s t a t i s t i c i s obtained when the sample i s divided between 1964 and 1965, where i t i s 2.554, but t h i s i s less than the 5% s i g n i f i c a n c e l e v e l of 3.52. Goldfeld and Quandt te s t s s i m i l a r l y give no evidence of s t r u c t u r a l break. A casual look at the data i n Table 2.1 reveals that the r e a l wage increased over the f i r s t part of the sample but seemed to l e v e l o f f thereafter. The OLS estimates given above suggest that i f the serie s i s stationary i t has an estimated mean of 4.58. Given a 1961 value of 2.62, i f one used the estimates we obtained of the parameters of the AR(1) regression, one would predict an increase i n the wage over some time followed by a l e v e l l i n g o f f , which perhaps provides the i n t u i t i o n behind why there seems to be no discernable pattern i n the re s i d u a l s . A time trend added to equation (3.2) proved to be i n s i g n i f i c a n t , with a t - s t a t i s t i c of only 0.548. Note that i n the models we estimate i n Chapters 3 and 4, we w i l l assume firms form expectations using these simple 3 autoregressive processes, so the residuals of these regressions t r a n s l a t e i n the models into forecast errors by firms. If expectations are to be described as r a t i o n a l i n the model, there should be no information embodied i n the residuals, and i t i s f o r t h i s reason we have examined the properties of the residuals of the r e n t a l rate and wage equations i n such depth. This i s standard p r a c t i c e i n empirical applications of these models; see a l l the papers r e f e r r e d to i n Chapter 1.4. Modelling input demands i n the manufacturing sector i s s i m p l i f i e d i f i t can be assumed i n the model that r e a l wages are not caused (in the Granger (1969) - Sims (1972) sense) by r e a l r e n t a l rates on c a p i t a l , the demand f o r c a p i t a l , or the demand f o r labour, and that r e a l r e n t a l rates are not caused by r e a l wages, the demand for c a p i t a l , or the demand for labour. This assumption greatly s i m p l i f i e s the s p e c i f i c a t i o n of the firms' input decision rules, because i t means that there i s no feedback from the firms' input decisions to input p r i c e s . The assumption i s used by Sargent (1978), Meese (1980), and Epstein and Yatchew (1985) among others to obtain a t r a c t a b l e s o l u t i o n . The technique used here (and i n the above mentioned papers by Meese and Epstein and Yatchew) f o r t e s t i n g the exogeneity of input prices i s from Geweke (1978) . We begin by estimating the vector autoregression s t " a + V t - i + V t - 2 + b t + e t ( 3 - 5 ) where S = (k , 1 , r , w. )', k i s c a p i t a l , 1 i s labour, r i s the r e a l r e n t a l rate of c a p i t a l , w i s the wage, a and b are 4x1 vectors of parameters, A^ and A^ are 4x4 matrices of parameters, t i s time, and i s a vector of errors, s e r i a l l y uncorrelated but perhaps c o r r e l a t e d across equations. We consider two a l t e r n a t i v e s to u n r e s t r i c t e d estimation of (3.5):(i) that the lagged k and 1 terms have zero c o e f f i c i e n t s i n the r and w equations, and ( i i ) that the lagged r and w terms have zero c o e f f i c i e n t s i n the k and 1 equations. The evidence i s that there i s stronger c a u s a l i t y from lagged p r i c e s to current input l e v e l s than there i s from lagged input l e v e l s to current p r i c e s . The l i k e l i h o o d r a t i o t e s t s t a t i s t i c f o r n u l l hypothesis (i) i s 18.225 and the Wald t e s t s t a t i s t i c i s 2 23.329 (each of which i s d i s t r i b u t e d x with 8 degrees of freedom). The l i k e l i h o o d r a t i o s t a t i s t i c f o r n u l l hypothesis ( i i ) i s 21.630 and the Wald te s t s t a t i s t i c i s 27.224. With the small sample we are using we cannot say with any confidence whether the c a u s a l i t y i s s t a t i s t i c a l l y s i g n i f i c a n t . Using the l i k e l i h o o d r a t i o and Wald te s t s t a t i s t i c s as given above leads one to r e j e c t the exogeneity of any of the variables i n question. But Epstein and Yatchew suggest modifying the l i k e l i h o o d r a t i o s t a t i s t i c s i n a way suggested by Nelson and Schwert (1982) by m u l t i p l y i n g the s t a t i s t i c s by (T - K) /T where T i s the sample si z e and K i s the number of parameters i n the u n r e s t r i c t e d model (2.7). This i s meant to correct f o r the problem of using large sample theory to examine a small-sample model. Since T = 22 and K = 40, the amended s t a t i s t i c i s meaningless for our purposes, or perhaps warns us that with t h i s small a sample we simply cannot say anything, i f the Nelson and Schwert co r r e c t i o n i s appropriate. 3.2 The Model The model i s c a l l e d l i n e a r quadratic because the quadratic s p e c i f i c a t i o n of both the adjustment costs and the output function leads to input demands which are l i n e a r i n r e a l input p r i c e s . An assumption of the model i s that firms form expectations of future input p r i c e s r a t i o n a l l y . Rational expectations are defined as expectations formed as a r e s u l t of using a v a i l a b l e information e f f i c i e n t l y . More s p e c i f i c a l l y i t means i n d i v i d u a l s and firms i n 48 the model make use of (a) past observations of variables and (b) knowledge of the structure of the economic model i n forming expectations. In t h i s model of input demand r a t i o n a l expectations are incorporated by assuming that firms know past values of r e a l input p r i c e s , and that an VAR(l) s p e c i f i c a t i o n can be used to represent the "model" firms use to make forecasts. See Chapter 1 fo r a survey of models of t h i s type that have been analysed and estimated. In t h i s chapter the estimation technique i s taken from Epstein and Yatchew (1985) . The assumptions and fun c t i o n a l form used i n t h e i r paper are not d i f f e r e n t from other l i n e a r quadratic r a t i o n a l expectations models of input demand, but the parameterization of the estimating equations i s d i f f e r e n t . Its advantage i s that the cross-equation r e s t r i c t i o n s that are implied i n r a t i o n a l expectations models of input demand are more simply s p e c i f i e d than i n other parameterizations using e s s e n t i a l l y i d e n t i c a l models (e.g. Hansen and Sargent (1980, 1981) on one hand, or Chow (1980b, 1981, 1983) on the other) . For ease of reference Epstein and Yatchew's notation i s used. A fi r m produces output y with inputs c a p i t a l k and labour 1. Define the vector x̂_ = (k^ 1^) ', and the production function i s y = a'x t + x t'Ax t/2 + (xfc - x t _ 1 ) ' B ( x t - x t_ 1)/2 + S1 {t), (3.6) where a i s a 2x1 vector of parameters, A and B are each 2x2 matrices of parameters, and Ŝ " (t) i s a sca l a r time trend meant to capture changes i n technology. Matrix A i s symmetric and negative d e f i n i t e and B i s diagonal and negative d e f i n i t e . Hansen and S a r g e n t (1981) and Chow (1980b, 1981) a l l o w n on-zero o f f - d i a g o n a l terms i n t h e a d j u s t m e n t c o s t m a t r i x B, b u t t h i s c a r r i e s t h e c o s t o f c o m p l i c a t i n g t h e s o l u t i o n o f t h e model c o n s i d e r a b l y . C o s t s o f a d j u s t i n g i n p u t l e v e l s a r e c a p t u r e d by t h e t e r m i n (3.6) i n v o l v i n g B. They a r e c a l l e d " i n t e r n a l a d j u s t m e n t c o s t s " because t h e c o s t s a r e e x p r e s s e d i n terms o f l o s t o u t p u t (see Treadway (1969, p . 2 2 9 ) ) . Note t h a t a d j u s t m e n t c o s t s depend on n e t changes i n i n p u t l e v e l s . I n G o u l d (1968) a d j u s t m e n t c o s t s a r e b a s e d on g r o s s i n v e s t m e n t , w h i l e i n Lucas (1967a) t h e y a r e b a s e d on t h e p e r c e n t a g e change i n i n p u t l e v e l s . One c o u l d i m a g i n e t h a t i n t h e model (3.1) a d j u s t m e n t c o s t s a r e c a p t u r i n g t h e d i s r u p t i o n i n v o l v e d i n c h a n g i n g t h e l e v e l o f any i n p u t , so t h a t no a d j u s t m e n t c o s t s a r i s e f r o m p u r e l y r e p l a c e m e n t i n v e s t m e n t . The r e a l r e n t a l r a t e o f c a p i t a l i s r and t h e r e a l wage i s ŵ _, and t h e v e c t o r o f i n p u t p r i c e s i s w r i t t e n ẑ _ = (r^_ ŵ _) ' . A t t i m e t = 0 a f i r m chooses a r u l e f o r s e t t i n g x^ t o s o l v e t h e p r o b l e m max E 0 ^ p t [ a ' x t + x ^ / 2 + (x f c - x ^ ' B ^ - x ^ ) / 2 + S 1 ( t ) - z f c'x ] (3.7) s u b j e c t t o x g i v e n , where p = (1 + R) and R i s a c o n s t a n t r a t e o f d i s c o u n t . Each t i m e p e r i o d t h e f i r m r e c a l c u l a t e s t h e s o l u t i o n t o t h e pr o b l e m , making use o f any new i n f o r m a t i o n . I n p u t p r i c e s f o l l o w t h e p r o c e s s (3.8) 50 where t h e 2x1 v e c t o r v and t h e 2x2 m a t r i x 0 a r e p a r a m e t e r s and e i s a random e r r o r t e r m . An advantage o f t h e l i n e a r q u a d r a t i c s p e c i f i c a t i o n i s t h a t t h e p r o b l e m (3.7) can be s o l v e d under t h e a s s u m p t i o n t h a t t h e f i r m has p e r f e c t f o r e s i g h t ; i . e . c e r t a i n t y e q u i v a l e n c e a p p l i e s . The s o l u t i o n has t h e f o r m x* - x t _ 1 = M<xt-1 - i t>, (3.9) where x f c i s t h e o p t i m a l d e c i s i o n a t t i m e t , x i s t h e " t a r g e t l e v e l " o f x a t t i m e t , and M i s t h e "a d j u s t m e n t m a t r i x " ( E p s t e i n and Yatchew (1985, pp. 239-40)). The m a t r i x M s o l v e s t h e e q u a t i o n 2 -1 -1 M - (1 + R)B AM - RM - B A ( l + R) =0, (3.10) and x i s g i v e n by x = A _ 1 ( J t - a ) , (3.11) where J = D Z (I + D ) " ( S t + ^ E z , t t s s=t (3.12) D = AB (1 + R) + R - M', (3.13) where R i s a 2x2 d i a g o n a l m a t r i x w i t h e v e r y d i a g o n a l e n t r y e q u a l t o R, and I i s a 2x2 i d e n t i t y m a t r i x . The v e c t o r J i s a w e i g h t e d average o f c u r r e n t and e x p e c t e d f u t u r e i n p u t p r i c e s . I f e x p e c t a t i o n s a r e s t a t i c , s ay t h a t E z = z t s f o r a l l s = t,...,oo, t h e n = z. W i t h r a t i o n a l e x p e c t a t i o n s , however, J i s c l e a r l y g o i n g t o depend somehow on c u r r e n t p r i c e s and on t h e p a r a m e t e r s o f t h e model u s e d t o f o r e c a s t f u t u r e p r i c e s , namely v and 6. The s o l u t i o n f o r J g i v e n by E p s t e i n and Yatchew (p. 241) i s as f o l l o w s : J f c = a + 0z t, (3.14) where a and £ a r e d e f i n e d by v = |3 X D a (3.15) and 0 = (3 1 ( ( I + D)|3 - D) (3.16) 4 and D i s as d e f i n e d i n (3.13). I n t h e f o l l o w i n g c h a p t e r t h e i n p u t p r i c e a u t o r e g r e s s i o n s a r e assumed t o be i n d e p e n d e n t f r o m one a n o t h e r ; i . e . 0 i s assumed t o be a d i a g o n a l m a t r i x . W h i l e one might t h i n k t h i s c o u l d be a u s e f u l s i m p l i f y i n g a s s u m p t i o n i n t h i s model, i n f a c t i t would g r e a t l y c o m p l i c a t e m a t t e r s , a d d i n g a number o f r e s t r i c t i o n s t o t h e e s t i m a t i o n . We do n o t know how much d i f f e r e n c e i n t h e r e s u l t s o f 52 As a f i n a l s t e p i n d e r i v i n g t h e e s t i m a t i n g e q u a t i o n s , d e f i n e P = BM. The s i m p l i f i e d p a r a m e t e r i z a t i o n o f E p s t e i n and Yatchew r e f e r e d t o e a r l i e r w i l l d e f i n e t h e i n p u t demand e q u a t i o n s i n terms of a, B, P, and J ^ . E q u a t i o n (3.10) can be s o l v e d f o r A as A = P / ( l + R) - B + B ( I + M) 1 . (3.17) This represents parameter r e s t r i c t i o n s on the s o l u t i o n to (3.7). Other r e s t r i c t i o n s derived i n Lucas (1967b) are M has 2 r e a l eigenvalues between -1 and 0 (3.18) and P i s symmetric and p o s i t i v e d e f i n i t e . (3.19) Writing the s o l u t i o n of the model i n the form i n which i t i s to be estimated we have x = (I + B ~ 1 P ) x t _ 1 - B _ 1 P A 1 ( J t - a) + u f c (3.20) and z t - v + 9 z t - i + v ( 3 - 2 1 ) the models of t h i s and the following chapter are due to t h i s d i f f e r e n t treatment of the evolution of input p r i c e s . 53 where A i s as defined i n (3.17) . The parameters to be estimated are B, P, a, a, and Technology i s completely described by B, P, and a, while a and /3 are the parameters r e l a t i n g v and S to input demands. In the estimation v and S are expressed i n terms of a and |3. R e s t r i c t i o n (3.17) i s imposed w r i t i n g (3.20) using B, P, and a as the only technology parameters. R e s t r i c t i o n (3.19) i s half-imposed, as P i s confined to be symmetric but i s not confined to be p o s i t i v e d e f i n i t e , and r e s t r i c t i o n (3.18) i s not imposed (but i s s a t i s f i e d by the data i n any case, as we see below). Error terms û_ are meant to r e f l e c t "random errors of optimization and errors i n the data" (Epstein and Yatchew p. 243) . In p r i n c i p l e they should be independent of the residuals e from (3.21), but t h i s r e s t r i c t i o n i s not imposed. Before the estimates are presented, i t i s i n t e r e s t i n g to see how Lucas' (1976) c r i t i q u e of econometric p o l i c y evaluation applies to the problem of input demands. A change i n the p o l i c y governing r e n t a l rates on c a p i t a l (e.g. a change i n corporate tax policy) would change the parameters of v and 8 . This i n turn, through (3.16) and (3.14), changes the parameters a and 0, which changes the r e l a t i o n between J and current input p r i c e s , which changes the parameters r e l a t i n g input prices to input demands. In sum, the reduced form parameters of the input demand equations change when v and 8 change. Lucas warns p o l i c y analysts to r e a l i z e that t h i s change occurs. 3.3 Estimates Epstein and Yatchew assume that firms can observe t h i s period's input p r i c e s before having to decide t h i s period's input l e v e l s . Chow (1980b, 1981), i n his formulation of an otherwise i d e n t i c a l model, assumes firms must choose period t input l e v e l s based only on observations of period t-1 (and e a r l i e r ) p r i c e s . Since here we are not c e r t a i n about the v a l i d i t y of the model, both s p e c i f i c a t i o n s w i l l be estimated. Tables 3.1 and 3.2 show the u n r e s t r i c t e d estimates of the reduced form of the 4 equation model (3.20) and (3.21), under the assumption that firms choose period t input l e v e l s a f t e r period t prices become known and under the assumption that the period t input l e v e l s must be chosen before period t prices become known, res p e c t i v e l y . The u n r e s t r i c t e d models are estimated with SHAZAM's (White (1978)) three-stage l e a s t squares. No parameter signs change across the two sets of estimates, although magnitudes change s l i g h t l y . Surprisingly, i n both tables the c o e f f i c i e n t on wages i n the labour equation i s p o s i t i v e , although not s i g n i f i c a n t l y so. R e s t r i c t e d estimates of the models are given i n Tables 3.3 and 3.4. In the tables the following notation i s used: P = "p p 1 2 B = ' B l °" a = V a = /3 = " P l l P12 P P 1-2 3-J •° B2- -a2- -a2- -P21 P22- The models are estimated with the non-linear maximum-likelihood option of SHAZAM. The data i s annual Canadian manufacturing from 1962 to 1984. S t a r t i n g values for the maximization process were chosen by f i r s t estimating a s t a t i c expectations version of the model. A s e l e c t i o n of d i f f e r e n t 55 s t a r t i n g values i n the neighbourhood of the f i n a l s o l u t i o n a l l converged on the estimates shown. Table 3.3 gives estimates under the assumption that firms are able to observe current input prices before s e t t i n g input l e v e l s . The implied reduced-form parameters are very d i f f e r e n t from the un r e s t r i c t e d estimates of Table 3.1. In p a r t i c u l a r , the demands fo r c a p i t a l and labour are each completely i n e l a s t i c with respect to both input p r i c e s ; the input demand equations reduce to a simple b i - v a r i a t e autoregression with constant terms. This s p e c i f i c a t i o n then leaves no p o s s i b i l i t y f o r any simulations of in t e r e s t regarding changes to the r e n t a l rate. Table 3.4 gives estimates under the assumption that firms must choose any year's input l e v e l s before p r i c e s are observed. Comparing these r e s u l t s to Table 3.3, we f i n d i n Table 3.4 that the demand f o r c a p i t a l does respond to r e n t a l rates, although the impact e l a s t i c i t y i s only approximately -.004, and i t does not i n Table 3.3, but otherwise the reduced forms are the same between the two cases. The estimates of s t r u c t u r a l parameters are very d i f f e r e n t i n magnitude across the two s p e c i f i c a t i o n s , although a l l are of the same sign across the two tables. But since M, the adjustment matrix, equals B ^P, the absolute values of B and P are not going to be well i d e n t i f i e d . Since the model of Table 3.4 i s the only one which could conceivably be of i n t e r e s t to one examining the e f f e c t s of r e n t a l rates on investment, the rest of t h i s chapter w i l l focus on t h i s model. The r e s t r i c t i o n (3.18), that M have 2 r e a l eigenvalues between 0 and -1, i s s a t i s f i e d , although i t was not imposed i n the estimation. Since M i s given by B 1P, the estimated value of M i s M -.1144 1.9576 ,0003 -.1883 and the eigenvalues of M are -.1237 and -.17 90. R e s t r i c t i o n s which are not s a t i s f i e d are that P be p o s i t i v e d e f i n i t e and that B be negative d e f i n i t e . In p a r t i c u l a r i s p o s i t i v e , which i s the wrong sign; i t suggests negative costs to adjusting the labour input. Also, consider the matrix A, from the production function, which should be negative d e f i n i t e to ensure constant or decreasing returns to scale. Its implied value has been given above; the p o s i t i v e element i n column 2 row 2 implies an increasing marginal product of labour. V i r t u a l l y a l l aspects of the "labour side" of t h i s model f a i l . Adjustment costs are the wrong sign, and the marginal product of labour at various data points indicate a negative marginal product that i s increasing. Yet on the " c a p i t a l side", adjustment costs are p o s i t i v e , and the marginal product of c a p i t a l i s found to be p o s i t i v e and decreasing. In the reduced form the r e s u l t i s a labour demand that i s completely i n e l a s t i c with respect to both input prices and to the c a p i t a l stock, and i s a c t u a l l y just an AR(1) process with a constant term, where the mean of the process i s estimated at 3654.7. So the s i g n i f i c a n t e f f e c t of labour input on c a p i t a l demand i n the c a p i t a l equation i s simply a term capturing t h i s AR process. It i s i n t e r e s t i n g to note that Epstein and Yatchew's r e s u l t s of t h i s same 4 equation model with r a t i o n a l expectations, using U.S. annual manufacturing data from 1948 to 1977, are very s i m i l a r to the re s u l t s obtained here (see t h e i r Table 5, p. 249). In both cases estimated signs are > 0, P^ and P^ < 0 (so neither case s a t i s f i e s the r e s t r i c t i o n P p o s i t i v e d e f i n i t e ) , and < 0 and B^ > 0 (so both cases have B^ being the wrong sig n ) . The marginal s i g n i f i c a n c e of the tes t of the cross equation r e s t r i c t i o n s i s found by taking the difference i n the log of the determinant of the sigma matrices, mult i p l y i n g t h i s by the number of observations (23), and comparing t h i s t e s t s t a t i s t i c with the 2 X d i s t r i b u t i o n with 3 degrees of freedom. The marginal s i g n i f i c a n c e of the r e s t r i c t i o n s i s only 2.53%, which would suggest r e j e c t i o n of the model by the data. The impact e l a s t i c i t i e s of input demands, as of 1975, are: short run e l a s t i c i t y with respect to: r w c a p i t a l -.004 +.004 labour +.000 +.000 Long run e l a s t i c i t i e s are found by applying equation (3.11), which describes how steady state demands change with respect to a change i n J . A permanent change i n input prices would be represented by a change i n J , which i s a weighted index of current and expected future input values of inputs at time t, given i n i s given by our estimate of Table 3.4 the implied estimate of 58 p r i c e s . The change i n the target by x^, with respect to a change A . From the estimates given i n A i s -.00005997 .002329 .002329 .9458 At 1975 c a p i t a l and labour input l e v e l s are 18715.3 and 3513.0, respectively, and the re n t a l rate and the wage rate are .10 6 and 4.00. In that same year the elements of J corresponding the r e n t a l rate and the wage are .113 and 4.134 (from equation (3.14) and the estimates of a and P given i n Table 3.4). Applying equation (3.11) we obtain the target l e v e l s of c a p i t a l and labour i n 1975 of 24925.5 and 3613.5. So at 1975 the long run e l a s t i c i t i e s of input demand, where t h i s means the r e l a t i v e change i n target input l e v e l s per r e l a t i v e change i n J , are: long run e l a s t i c i t y with respect to: r w c a p i t a l -.069 + .006 labour + .001 + .001 As long as M i s a stable matrix, i . e . has 2 r e a l eigenvalues between -1 and 0, and our estimate i s a stable matrix, then the method of c a l c u l a t i n g long-run e l a s t i c i t i e s by examining the r e l a t i v e change i n target values i s the same as we would f i n d i f we looked at the r e l a t i v e change i n long run actual values of inputs. Consider the following. If -1 -1 xfc = (I + M)x f c_ 1 + MA a - MA J f c , (3.22) then s+1 2 x^, = ( I + M ) x ^ + ( I + ( I + M ) + ( I + M ) +...+ t+s t-1 s -1 -1 -1 ( I + M ) ) M A a - M A J - (I + M) MA J , -t+s t+s-1 ... - (I + M) SMA - 1J t. (3.23) If we change a l l J , T = t, t+1, ...,co, by AJ, then the change i n x i s t+s 2 s -1 Ax^ , = - ( 1 + ( I + M ) + ( I + M ) + ... + ( I + M ) ) MA A J, t+s (3.24) and as s co, i f M i s stable, t h i s converges to Ax^ , = - ( I - (I + M))" 1MA~ 1AJ = A _ 1 A J . (3.25) t+s So A ^ gives us the change i n the target input l e v e l s and the change i n the actual l e v e l s i f M i s a stable matrix. It i s i n t e r e s t i n g to note how d i f f e r e n t are the estimates of long run e l a s t i c i t i e s i f we ignore our s p e c i f i c a t i o n of expectations and simply take the reduced form parameters of input 60 demand as given f o r any expectations. Write the reduced form as x,. = (I + M)x̂ _ + r z , + c. (3.26) t t-1 t-1 Ignoring expectations, a change i n current and future input p r i c e s by Az would lead to a change i n x of t"t"S A x t + s = (I + (I + M) + (I + M) 2 + ... + (I + M) S)TAz, (3.27) and as s -> oo, i f M i s stable t h i s converges to Ax^, = -M 1 r A z . (3.28) t+s Taking the reduced form estimates of M and T from Table 3.4 the estimated long run e l a s t i c i t i e s are long run e l a s t i c i t y with respect to: r w c a p i t a l -.035 +.039 labour +.001 +.001 The estimate of the long run e l a s t i c i t y of c a p i t a l stock with respect to the r e n t a l rate, when estimated considering only the reduced form of the model, i s only about one hal f the s i z e of the estimate when we account f o r expectations as s p e c i f i e d i n equation (3.21). We cannot measure whether t h i s d i f f e r e n c e i n the estimates o f l o n g r u n e l a s t i c i t i e s i s s i g n i f i c a n t i n an e c o n o m e t r i c sense, s i n c e we have no measure o f s t a n d a r d e r r o r s . The l o n g r u n e l a s t i c i t y o f c a p i t a l w i t h r e s p e c t t o t h e r e n t a l r a t e t h a t we d o u b l e when t r e a t i n g e x p e c t a t i o n s as r a t i o n a l i s a v e r y s m a l l number ( i n e l a s t i c i t y t e r m s ) ; whether we s h o u l d r e g a r d t h i s d i f f e r e n c e i n e s t i m a t e s as q u a n t i t a t i v e l y i m p o r t a n t i s perhaps answered i n t h e f o l l o w i n g s e c t i o n o f t h i s c h a p t e r . There s i m u l a t i o n s o f t h e model a r e r u n under b o t h methods; where r e d u c e d forms a r e changed t o acc o u n t f o r a new regime, and where t h e y a r e n o t . Because b o t h c a p i t a l and l a b o u r a r e sl o w t o a d j u s t t o t h e i r t a r g e t l e v e l s , i t i s u s e f u l t o know what t h e medium r u n 2 e l a s t i c i t i e s a r e . The m a t r i x ( I + (I + M) + ( I + M) + . . .) i s slow t o c o n v e r g e t o -M . From o u r e s t i m a t e o f M o f -.1144 1.9576 M = -.0003 -.1883 2 s we f i n d t h a t (1+ ( I + M ) + ( I + M ) + ... + ( I + M ) ) e q u a l s 4.5158 19.3905 when s -.0030 3.7838 62 6.4027 44.0446 -.0068 4.7400 when s = 10, and that -M V-l .5093 88.4639 -.0136 5.1697 which indicates that convergence to long run l e v e l s i s very slow. Based on these c a l c u l a t i o n s , the f i v e - and ten-year e l a s t i c i t i e s , expressed i n terms of target l e v e l s so that they are more e a s i l y compared to the long run e l a s t i c i t i e s given above, are five-year e l a s t i c i t y ten-year e l a s t i c i t y r w r w c a p i t a l -.037 +.002 -.052 +.004 labour +.000 +.001 +.001 +.001 We also estimated the model with s t a t i c expectations. This i s achieved by s e t t i n g J^, which i s l i k e an index of current and expected future input p r i c e s , equal to z^._^ (or, equivalently, by se t t i n g a = 0 and |3 equal to the i d e n t i t y matrix) . The two equations of (3.20) are then estimated. Estimates of s t r u c t u r a l parameters under s t a t i c expectations are s i m i l a r to those f o r 63 dynamic expectations l i s t e d i n Table 3.4; no signs change, and r e l a t i v e magnitudes are roughly the same. S t a t i c expectations estimates are: P .14155xl0~ 3 (.00010) P 3 -2.6673 (.92691) B 2 12.536 (9.1424) a„ -3471.0 (1.4765) P -.22744xl0~ 2 (.00173) 2 -2 B -.11599x10 (.00086) a, -3.5094 (2.9434) This implies a reduced form f o r input demands of: k = -3664.545 + .878k , + 1.9611 , - 3725.996r , + 2.711w , t t - 1 t-1 t - 1 t - 1 1 = 770.645 - 0k t_ 1 + •7871 f c_ 1 + 2.711r t_ 1 + .219wfc_1 which i s s i m i l a r to those obtained with dynamic expectations (compare with Table 3.4), although the estimate of the e l a s t i c i t y of c a p i t a l demand with respect to the re n t a l rate i s much greater. The l i k e l i h o o d r a t i o s t a t i s t i c f o r the 2 equation s t a t i c expectations model against the u n r e s t r i c t e d 2 equation model i s 2 8.670, which i s d i s t r i b u t e d % with 3 degrees of freedom. The marginal s i g n i f i c a n c e of the r e s t r i c t i o n s i s then 3.36%. F i n a l l y , we turn to estimates of the production function, given i n equation (3.1). U n r e s t r i c t e d estimation y i e l d s 64 y = 29417 - 2.0855k + 2.92971 - .0001k2 - .002012 + .1322k 1 I* ^ ^ L* ^ ^ (126090) (4.5320) (91.362) (.00004) (.0166) (.0016) + .0001 (k t - \ _ 1 ) 2 ~ .0026 (1 - 1 t _ 1 ) 2 + 3238.8t. (.0012) (.0118) (815.16) The Wald x s t a t i s t i c f o r the j o i n t t e s t of both adjustment terms being zero i s .0747 with 2 degrees of freedom, suggesting i n s i g n i f i c a n c e . The signs on the c o e f f i c i e n t s lend further doubt on the usefulness of t h i s model. R e s t r i c t e d estimation of the output equation together with the four equation model of input demands and input prices y i e l d e d r e s u l t s with such a poor f i t of the data that they are not worth reporting. 3.4 Simulations Table 3.5 gives the r e s u l t s of a number of simulations made using the parameter estimates of the 4 equation r e s t r i c t e d l inear-quadratic model presented i n Table 3.4. Although the merits of t h i s model as an explanation of the data have been found to be dubious, the simulations at least i l l u s t r a t e the p r i n c i p l e s behind doing p o l i c y analysis with a r a t i o n a l expectations model, and i l l u s t r a t e the empirical s i g n i f i c a n c e of how expectations are s p e c i f i e d . For a l l the simulations i n Table 3.5, we imagine someone i n 1975 making long-range forecasts of the c a p i t a l stock (since labour i n these estimates seems to simply follow a predetermined path, the forecasts of labour are not recorded i n the table; a l l simulations lead to a forecast value of labour i n 1984 of 3603, 65 while i t s actual value turned out to be 3621) . Column A l i s t s the actual data. Column B l i s t s a forecast of c a p i t a l made i n 1975, using the model of Table 3.4, and gives the standard errors of the forecasts. The standard errors were found using the method given by Judge et. a l . (1988, pp. 764-67) f o r c a l c u l a t i n g the variance of forecasts with VAR(l) systems. The simulation i n Column C i s the r e s u l t of a negative shock to the r e n t a l rate on c a p i t a l i n 1975 by a f a c t o r of 10%. This lowers r i n 1975 from .1060 to .0954. The underlying time-series parameters of r are l e f t unchanged. This shock has two e f f e c t s on the path of c a p i t a l . F i r s t , the future r e n t a l rate depends on i t s past values, so even though the time-series parameters are unchanged, there w i l l be some pe r s i s t e n t e f f e c t s on r e n t a l rates from t h i s one-time shock. Second, since c a p i t a l demand responds to the shock to the r e n t a l rate i n 1976, and c a p i t a l demand depends on i t s own past values, there w i l l be further p e r s i s t e n t e f f e c t s . But given the s t a t i o n a r i t y of r e n t a l rates, and the fact that the adjustment matrix M i s "stable" (two r e a l eigenvalues between 0 and -1) , the e f f e c t of t h i s shock i n the very long-term tends asymptotically to zero. Comparing Columns B and C we f i n d the impact e f f e c t , i n 1976, i s to increase the c a p i t a l stock by .04% over what i t otherwise would have been. By 1984, the e f f e c t of the shock i s a c a p i t a l stock only .02% greater than what i t otherwise would have been. Eventually the e f f e c t s of the shock die out completely. The simulation i n Column D i s somewhat unusual. Here i n 1975 there i s a permanent decrease i n r e n t a l rates of 10%, but the path of wages i s l e f t unchanged. This i s achieved by lowering the 1975 66 value of the r e n t a l rate by 10% d i r e c t l y , as we d i d i n simulation C, and i n changing the b i v a r i a t e autoregressive process of the ren t a l rate and wages from Z t = v l " + " 911 9 1 2 ' •V2- -921 922- t-1 to z = t 9v„ 2-" 611 -"12 1 - 1 1 G 2 1 922 t-1' What i s unusual about the simulation i s not that there i s a permanent s h i f t i n the path of r e n t a l rates, but rather that we assume firms are unaware that the s h i f t i n the path has taken place. They observe r e n t a l rates c o r r e c t l y , but they do not r e a l i z e the change i n regime; i n t h e i r minds each year brings a s u r p r i s i n g l y low r e n t a l rate. This simulation i s presented as a contrast to the one i n Column E, which has the same permanent lowering of r e n t a l rates as simulation D, but which presumes that firms do r e a l i z e (immediately) that there has been a change i n regime, although they had not an t i c i p a t e d t h i s change at a l l , and reset t h e i r input demand rules accordingly. Estimating the revised input demand equations proceeds as follows. F i r s t , the change i n the path of r e n t a l rates has involved a change i n the values of the parameters v and 0 (see equation (3.21)). According to (3.20), input demands depend on input prices through the s t r u c t u r a l parameters B, P, and A, which 67 do not change with the changes i n v and 0, and through the parameters i n J^, which w i l l change with the changes i n v and 0. The r e l a t i o n s h i p between J and v and 0 i s given i n equations ( 3 . 1 4 ) , ( 3 . 1 5 ) , and ( 3 . 1 6 ) . With the new values of v and 0, new values of a and 0 are implied. The new values of a and /3 are a = .0995 . 6430 .3965 .5909 .0094 ,8836 Comparing these values of a and 0 to Table 3.4 we see, as we would expect, no r a d i c a l changes. The second step i s to incorporate the new a and 0 into the input demand equations. The new input demand equations w i l l i n the reduced form have d i f f e r e n t c o e f f i c i e n t s on the input p r i c e terms, and the constant terms w i l l also change. The reduced form parameters r e l a t i n g current input demand to the previous year's l e v e l s do not change, as they depend only on the s t r u c t u r a l parameters P and B. The new reduced form of the system i s k t .886 1.958 -720. 808 19.167 k t - l -4206 .004 0 .812 • 837 .148 V i + 687 .044 r t 0 0 • 670 -.006 r t - l .060 w t 0 0 1. 587 .981 w t - l - .039 and the simulation i n Column E i s based on t h i s system. 68 The change i n the input demand equations has a sub s t a n t i a l e f f e c t on the r e s u l t s . Compare Columns B, D, and E. Columns D and E involve the same lowering of r e n t a l rates. For 1976 simulation D gives a c a p i t a l stock .04% higher than i t would otherwise have been, but simulation E has a c a p i t a l stock that i s .12% higher than i t would otherwise have been. The values f o r 1980 are D: .18% higher and E: .48% higher. For 1984 we have D: .26% and E: .67%. Even though i n t h i s model c a p i t a l i s quite i n e l a s t i c with respect to r e n t a l rates, we f i n d that accounting f o r the changes i n input demand rules that should take place i f the change i n the re n t a l rate path i s recognized by the f i r m and incorporated into t h e i r input demand rules leads to a differ e n c e i n the predicted e f f e c t s of the r e n t a l rate change by a factor of around 2 or 3. While one might i n t e r p r e t t h i s r e s u l t as suggesting that how we specif y the expectations process can have large e f f e c t s , we must keep i n mind that t h i s p a r t i c u l a r model was rejected by the data, and that we should not form general conclusions based on the re s u l t s of t h i s chapter. 3.5 Conclusions The l i n e a r quadratic r a t i o n a l expectations model of input demand, estimated with Canadian manufacturing data, i s found wanting i n many respects. Some r e s t r i c t i o n s implied by the model were accepted by the data, others were not. The model generated a demand for c a p i t a l equation close to that obtained by u n r e s t r i c t e d regression, although the r e s t r i c t i o n s reduced the e l a s t i c i t y of c a p i t a l with respect to r e n t a l rates. But a demand for labour that i s p e r f e c t l y i n e l a s t i c with respect to both input prices must be somewhat suspect. Also, the adjustment costs f o r labour were of the wrong sign. In the following chapter we estimate an a l t e r n a t i v e model of dynamic input demand, and compare the performance of the two models. 70 TABLE 3.1 Estimate of Unrest r i c t e d Linear-Quadratic Model where Input Demands Depend on Current Input P r i c e s . Dependent va r i a b l e k Dependent variable 1 k t - l .89175 (.02120) a -.01311 (.01080) V i 1.8416 (.24982) .67065 (.12288) r t -5814.4 (3724.3) -2387.2 (2347.6) w t -93.470 (179.06) 167.51 (107.49) constant -2852 .3 (1035.1) 1066.6 (625.19) standard err o r 173.08 115.97 Dependent va r i a b l e r Dependent va r i a b l e w r t - l .71315 (.13746) .95233 (1.8451) W t - 1 -.00588 (.00411) .93761 (.05524) constant .05832 (.02916) .17842 (.39143) standard error .00908 .12193 sigma = 29958. 8869.3 .03047 -.36046 13449. .10051 •9.4383 .00008 .00021 .01487 log of determinant of sigma = 5.0322 a Standard errors of parameter estimates i n parentheses, b sigma i s the variance-covariance matrix of the 4 equation system where the order of the equations i s , by dependent var i a b l e , k, 1, r, w. 71 TABLE 3.2 Estimate of Unre s t r i c t e d Linear-Quadratic Model where Input Demands Depend on Lagged Input P r i c e s . Dependent va r i a b l e k Dependent variable 1 k t - l .89080 (.02172) a -.01246 (.01049) V i 1.8431 (.24106) .65938 (.11636) r t - l -4295.1 (3073.8) -1542.7 (1677.4) w t - l -47.842 (154.29) 170 .79 (78.072) constant -3210.4 (860.60) 986.56 (447.35) standard error 181.02 102.49 Dependent va r i a b l e r^ Dependent v a r i a b l e w r .71315 (.13746) .95233 (1.8451) w -.00588 (.00411) .93761 (.05524) constant .05832 (.02916) .17842 (.39143) standard e r r o r .00908 .12193 sigma = 32769. 9783.5 -.46505 -2.9321 10504. -.06417 -9.4383 .00008 .00021 .01487 log of determinant of sigma = 5.0314 a Standard errors of parameter estimates i n parentheses, b sigma i s the variance-covariance matrix of the 4 equation system where the order of the equations i s , by dependent variable, k, 1, r, w. 72 TABLE 3.3 Estimate of R e s t r i c t e d Linear-Quadratic Model where Input Demands depend on Current Input Prices Technological Parameters P 18.449 ( 2 0 0 1 . 7 ) a P 2 -320.94 (34821.) P 3 -65630. (7120600.) B 1 -166.35 (18048.) B 2 298440. (32379000.) a -522700. (56711000.) a 2 -87360000. (9478100000.) Parameters of J a .11469 (.02207) (S .37882 (.08566) jS -.82642 (1.6725) a 2 .52768 (.40223) 0 2 -.01073 (.00454) (S-. .92099 (.07531) b sigma 36060. 10874. -.50141 -2.9683 13923. -.10572 -9.5567 .00008 .00020 .01568 log of determinant of sigma = 5.4362 marginal s i g n i f i c a n c e of r e s t r i c t i o n s = .0257 R e s t r i c t e d estimates of reduced form k = -4207.088 + .889k , + 1.9291^ , - .010r + 0w^ t t-1 t - 1 t t 1 = 810.933 - .001k + .7801 + Or + Ow Iv *V Iv \~ Iv r = .071 + . 6 5 2 r t _ 1 - .007w f c_ 1 w =-.084 + 1.382r , + .994w^ , a Standard errors of parameter estimates i n parentheses. b sigma i s the variance-covariance matrix of the 4 equation system where the order of the equations i s , by dependent variable, k, 1, r, w. 73 TABLE 3.4 Estimate of R e s t r i c t e d Linear-Quadratic Model where Input Demands depend on Lagged Input Prices Technological Parameters P .28307x10 (.01858)3 P -.48440x10 (.32678) -2 P 3 -2.9801 (38.030) B 1 -.24745x10 (.16848) B 2 15.830 (197.64) a -6.8069 (464.49) a 2 -3470.9 (46354) Parameters of J a .11043 (.12705) 0 .39786 (.75508) 0 -.79212 (3.0780) a 2 .68720 (2.2692) 0 -.00993 (.01935) /3 2 2 .88276 (.51155) 35224. 10487. 13861. -.46782 -.10919 .00008 -2.7036 -9.2650 .00021 .01530 log of determinant of sigma = 5.4367 marginal s i g n i f i c a n c e of r e s t r i c t i o n s = .0253 Re s t r i c t e d estimates of reduced form b sigma = k = -4225.810 + .886k , + 1.9581,. , - 723.758r , + 20.140w , t t-1 t-1 t-1 t-1 1 = 687.078 - 0k + .8121 , + .801r , + .147w , t t-1 t-1 t-1 t-1 r = .067 + •670r t_ 1 - •007wt_1 w = -.039 + 1.428r + .981w a Standard errors of parameter estimates i n parentheses, b sigma i s the variance-covariance matrix of the 4 equation system where the order of the equations i s , by dependent variable, k, 1, r, w. 74 TABLE 3.5 Forecast values of the c a p i t a l stock under various conditions. Simulation Year A B C D E 1975 18715 .3 18715 .3 (-) 18715 .3 18715 .3 18715 .3 1976 19404 .4 19229 .2 (187.7) 19236 .9 19236 . 9 19253 .2 1977 20004 .6 19721 .8 (390.5) 19733 .4 19737 .6 19768 .1 1978 20376 .6 20189 .1 (616.9) 20202 .2 20212 .7 20255 .8 1979 21013 .3 20628 .6 (839.0) 20641 .7 20659 .7 20713 .7 1980 21982 .7 21039 .1 (1044.5) 21051 .2 21076 .9 21140 .7 1981 23286 .0 21420 .1 (1228.7) 21430 .7 21464 .1 21536 .3 1982 23805 .9 21772 .0 (1390.3) 21780 .8 21821 .4 21901 .0 1983 23698 .8 22095 .4 (1529.8) 22102 .5 22149 .6 22235 .7 1984 23603 .4 22391 .6 (1648.9) 22396 .9 22450 .0 22541 .7 Description of Simulations A: Actual data f o r c a p i t a l stock (machinery and equipment). B: Simulated forecast using l i n e a r quadratic model, with reduced form estimates given i n Table 3.4, s t a r t i n g at 1975, standard errors i n parentheses. C: Simulated forecast, using the model of B, with a one-off negative shock to the r e a l r e n t a l rate of c a p i t a l i n 1975 of 10%. D: Simulated forecast, using the model of B, with a permanent lowering of the path of the r e a l r e n t a l rate by 10%, beginning i n 197 5, where firms do not r e a l i z e there has been a change i n regime. E: Simulated forecast, with a permanent lowering of the path of the r e a l r e n t a l rate by 10%, beginning i n 1975, where the reduced form parameters of the l i n e a r quadratic model have been adjusted to r e f l e c t the change i n the path of r e n t a l rates ( i . e . where firms do r e a l i z e there has been a change i n regime) CHAPTER 4 Epstein and Denny's Model An a l t e r n a t i v e model of input demand to that described and estimated i n Chapter 3, the model of Epstein and Denny (1983), i s estimated i n t h i s chapter, using the same data from the Canadian manufacturing sector from 1962 to 1984. A value function i s s a i d to have a f l e x i b l e f u n c t i o n a l form when i t provides a second order approximation to an a r b i t r a r y function that i s consistent with the underlying economic theory. 1 In t h i s chapter the value of the firm's cost minimization problem i s described by a f l e x i b l e f u n c t i o n a l form. Input demand functions are derived for the case of firms having s t a t i c expectations of input p r i c e s and the case of firms' forecasts of input p r i c e s being described by f i r s t order autoregressive processes. The f l e x i b l e f u n c t i o n a l form model of t h i s chapter i s s i m i l a r to the l i n e a r quadratic model of the previous chapter i n a number of respects. In both models firms use c a p i t a l and labour to produce a s i n g l e output, r e a l input prices are exogenous to the manufacturing sector, and there are i n t e r n a l , convex costs of adjusting the l e v e l s of inputs. In both models we assume firms forecast future input p r i c e s r a t i o n a l l y , where these r a t i o n a l forecasts are approximated i n the estimation with those generated by f i r s t order autoregressions (although a difference between the models i s that the l i n e a r quadratic model allows expectations to be modelled as higher order autoregressions i f desired, whereas See Diewert (1974, p. 133) or Epstein (1981, p. 87) t h e model i n t h i s c h a p t e r does n o t ) . One d i f f e r e n c e between t h e two models i s t h a t w i t h a f l e x i b l e f u n c t i o n a l form, by d e f i n i t i o n , t h e o n l y r e s t r i c t i o n s p l a c e d on ( t h e t e c h n o l o g y a r e t h o s e r e q u i r e d by t h e a s s u m p t i o n o f p r o f i t m a x i m i z a t i o n ( o r c o s t m i n i m i z a t i o n ) . T h i s a p p l i e s b o t h t o t h e t e c h n o l o g y o f " g r o s s o u t p u t " ( i . e . o u t p u t b e f o r e a d j u s t m e n t c o s t s have been s u b t r a c t e d ) and t o t h e s p e c i f i c a t i o n o f t h e a d j u s t m e n t c o s t s t h e m s e l v e s . F o r example, i n t h e l i n e a r q u a d r a t i c model a d j u s t m e n t c o s t s were assumed t o be q u a d r a t i c and a d d i t i v e , whereas i n t h e f l e x i b l e f u n c t i o n a l form, t h e more g e n e r a l s p e c i f i c a t i o n t h a t c o s t s be i n c r e a s i n g and convex w i t h r e s p e c t t o changes i n i n p u t s , and n o t n e c e s s a r i l y a d d i t i v e l y s e p a r a b l e , i s used. The o t h e r major d i f f e r e n c e i s t h a t i n t h e model o f t h i s c h a p t e r f i r m s t a k e t h e l e v e l o f o u t p u t as g i v e n a t any p o i n t i n t i m e , w h i l e i n t h e l i n e a r q u a d r a t i c model o u t p u t was endogenous. To o u r knowledge t h e r e a r e no e x i s t i n g models o f t h e f i r m t h a t have been e m p i r i c a l l y implemented where o u t p u t i s endogenous, e x p e c t a t i o n s a r e r a t i o n a l , t h e r e a r e a d j u s t m e n t c o s t s , and t h e 2 f u n c t i o n a l f o r m i s f l e x i b l e . We see below t h a t t r e a t i n g o u t p u t as g i v e n g r e a t l y i mproves t h e f i t o f t h i s model, r e l a t i v e t o t h a t o f t h e l i n e a r q u a d r a t i c model. F i n a l l y , t h e model o f t h i s c h a p t e r i m p l i c i t l y a l l o w s f o r t h e This would c e r t a i n l y be a worthwhile project. Epstein (1981) provides the t h e o r e t i c a l model f o r the case of s t a t i c expectations. contribution of a material input, while the l i n e a r quadratic model had only the two inputs, labour and c a p i t a l . 4.1 The Model The s p e c i f i c a t i o n i s taken d i r e c t l y from Epstein and Denny (1983) without a l t e r a t i o n . As i n Chapter 3 denote output at time t y , inputs x = (k 1.)', r e a l input p r i c e s z = (r w )', and the constant discount rate R. Input p r i c e s i n t h i s chapter are d e f l a t e d by the materials p r i c e index m, given i n Table 2.1, so z f c i n t h i s chapter i s very close to, but not i d e n t i c a l to, from Chapter 3. Both c a p i t a l and labour are quasi-fixed. Define a purely v a r i a b l e input, materials, v̂ _, to whose p r i c e we w i l l normalize the quasi-fixed input p r i c e s . The technology of a f i r m i s given by y t = F ( V V x t - V i 1 ' ^-^ The v a r i a b l e cost function i s given by c ( y f V x t " x t - i > = m i n { v t : y t - F ( V V x t - x t - i ) K v t (4.2) We assume for now that the f i r m expects current input p r i c e s and output to remain constant, although t h i s w i l l l a t e r be relaxed. Under s t a t i c expectations we set the firm's problem at time 0 as that of choosing a time path of input l e v e l s i n order to minimize over an i n f i n i t e horizon the present discounted value of future costs: 78 00 -Rt rain X e [C(y, x , x ) + z'x ]dt (4.3) • 0 X t subject to x = x - x , x given, and x > 0 for a l l t . *-r L> L* J - \ J L. Each time period expectations of input prices and output are revised, and the s o l u t i o n to problem (4.3) i s recalculated. Let V ( X g , y, z) be the value of the problem (4.3). We note here that below technology w i l l be defined i n terms of the form of V. If we define V and V as 2x2 and 2x1 matrices of p a r t i a l zx z d e r i v a t i v e s , respectively, then the optimal decision rule for adjusting x, as derived by Epstein (1981) , i s x*(x , y, z) = V ' 1 (x , y, z)[RV (x , y, z) - x ]. (4.4) t t zx t z t t Epstein and Denny (1983, pp. 651-2) l i s t the properties V 3 must s a t i s f y i f C i s to s a t i s f y c e r t a i n r e g u l a r i t y conditions. A p a r t i c u l a r s p e c i f i c a t i o n of V that s a t i s f i e s those properties i s There are s i x conditions: C must be p o s i t i v e , C must be increasing i n y and x and decreasing i n x, C must be convex i n x, a unique s o l u t i o n to problem (4.3) must e x i s t for each (x^, y, z), the unique s o l u t i o n must have a unique steady state input l e v e l x that i s g l o b a l l y stable, and for any ( xQ? YI x ) there e x i s t s a * . * vector of input p r i c e s z such that x i s the optimal p o l i c y at time 0 i n problem (4.3) given (x , y, z ). 79 V(x , y, z) = [z' 1] y/2 + ( z ' * " 1 + A')x x t (4.5) + R 1 ( z / * 1A + h) + Q'x /y + x'Q x /2y x t t xx t where $, and Q are each 2x2 matrices of parameters, <f>, A , A, X X X and are each 2x1 vectors of parameters, and b and h are scalar parameters. Combining the s o l u t i o n (4.4) of the dynamic problem (4.3) with the f l e x i b l e f u n c t i o n a l form (4.5), the optimal rule f o r xfc i s X t = M < X t - l " X ) ' (4.6) where M = R - *, (4.7) x(y, z) = - (R - tf) _ 1{R* [$z + <P]y + A}, (4.8) and where R i s (as i n Chapter 3) a diagonal matrix with each diagonal element equal to R. The vector x represents the steady state, or target, demands f o r the quasi-fixed f a c t o r s , and i s a function of the l e v e l of output and of input p r i c e s . The optimal rule (4.6) has a reduced form i d e n t i c a l to that which a r i s e s i n the l i n e a r quadratic model (see equation (3.9)). The s t r u c t u r a l parameters underlying M are c l e a r l y d i f f e r e n t across the two models, however. 80 The r e s t r i c t i o n s which V must s a t i s f y to be consistent with cost minimization, and which w i l l be tested i n the estimation of the model are $ i s symmetric (4.9) M has 2 r e a l eigenvalues between -1 and 0. (4.10) Condition (4.10) was also imposed on the l i n e a r quadratic model (see condition (3.13)). The input demand functions implied by (4.6), (4.7), and (4.8) w i l l be estimated. A f i n a l step before estimating the model w i l l be to incorporate t e c h n i c a l change by changing (4.1) to y t = e y t(m t, xfc, x f c - x ^ ) , (4.11) where j represents the exponential rate of technological change. -yt In the analysis above substitute y^e for y^. Defining the following reduced form parameters using the same notation as Epstein and Denny, E = R m , (4.12) G = Rtf#, (4.13) the estimating equation f o r t h i s s t a t i c expectations case i s 81 x t / y t = ( i + M ) x t _ i / y t + [ E z t + G ] ( 1 + y ) _ t + x / y t + u t ' (4.14) where i s a random e r r o r v e c t o r . E p s t e i n and Denny use t h e a s s u m p t i o n t h a t changes i n x a f f e c t y o n l y a f t e r a one p e r i o d l a g , so y i s p r e d e t e r m i n e d i n ( 4 . 1 4 ) . T h i s a l l o w s t h e use o f s t a n d a r d e c o n o m e t r i c t e c h n i q u e s . Below, when t h e model i s e s t i m a t e d w i t h n o n - s t a t i c e x p e c t a t i o n s , we c o n s i d e r as a l t e r n a t i v e s ( i ) i n p u t demands d e p e n d i n g on l a g g e d o u t p u t , and ( i i ) i n p u t demands d e p e n d i n g on p r e d i c t e d c u r r e n t o u t p u t where t h e p r e d i c t i o n i s made i n t h e p r e c e d i n g t i m e p e r i o d . We assume i n p u t p r i c e s a r e exogenous, a l t h o u g h we f o u n d i n C h a p t e r 3 t h a t w i t h t h i s d a t a s e t t h i s a s s u m p t i o n may be s u s p e c t . However, e s p e c i a l l y when we work w i t h n o n - s t a t i c e x p e c t a t i o n s i n t h i s model, i t i s an a s s u m p t i o n w h i c h must be made f o r p u r p o s e s o f e s t i m a t i o n . 4.2 Estimates E q u a t i o n (4.14) i s e s t i m a t e d w i t h t h e n o n - l i n e a r maximum l i k e l i h o o d o p t i o n o f SHAZAM. T a b l e 4.1 g i v e s t h e r e s u l t s of e s t i m a t i o n of (4.14) w i t h t h e r e s t r i c t i o n (4.9) imposed. The n o t a t i o n i s * = 11 12 21 22 $ $ 11 12 $ $ 21 22' r <P = ' V J •*2 • • \ - The s i n g l e r e s t r i c t i o n imposed on t h e e s t i m a t e s , w h i c h i s = <$2l' i s r e j e c t e d . The o t h e r r e s t r i c t i o n imposed by t h e model i s t h a t M be a s t a b l e a d j u s t m e n t m a t r i x . S i n c e t h e c h o i c e o f a v a l u e o f t h e d i s c o u n t r a t e , R, does not a f f e c t t h e r e d u c e d f o r m o f t h e model, we f o l l o w E p s t e i n and Denny by s e t t i n g R = .07. Then t h e 82 estimated value of M i s -.1212 1.5230" F .0420 -.9026_ which has eigenvalues of -.047 and -.977, s a t i s f y i n g r e s t r i c t i o n (4.10) . Epstein and Denny (p. 659) i n t e r p r e t the elements of M as follows. If labour i s at i t s steady state value, 12% of the adjustment of the c a p i t a l stock towards i t s steady state l e v e l occurs i n one year. If c a p i t a l i s at i t s steady state l e v e l , 90% of the adjustment i n labour occurs i n one year. Using U.S. manufacturing data f o r the annual observations 1947 to 1976 Epstein and Denny obtain the i d e n t i c a l adjustment parameters of 12% and 90%. In the l i n e a r quadratic model of Chapter 3, the respective rates of adjustment were 11% and 21%. So for some reason the l i n e a r quadratic s p e c i f i c a t i o n predicts a much slower rate of adjustment f o r labour than does the more f l e x i b l e s p e c i f i c a t i o n , but the two models each predict the same speed of adjustment for c a p i t a l . The reduced form parameters implied by the r e s t r i c t e d estimation are A " -.2012 .0030' A .0206 A ' -2890.4 E = , G = , and \ -.0140 -.0092 .0899 319.82 The diagonal elements of E are the own-price c o e f f i c i e n t s f o r the capital/output and labour/output r a t i o s , so t h e i r negative sign i s expected. The impact e l a s t i c i t i e s generated by the s t a t i c expectations M = 83 model are, c a l c u l a t e d at 1975 l e v e l s , as follows: short run e l a s t i c i t y with respect to: r w Y c a p i t a l -.048 + .027 + .026 labour -.018 -.439 + .615 The short run e l a s t i c i t i e s are very s i m i l a r to those obtained by Epstein and Denny (1983, p. 661), i n p a r t i c u l a r the own-price e l a s t i c i t i e s (although our estimate i s for a labour demand more e l a s t i c with respect to the wage than they obtain) and the output e l a s t i c i t i e s . To c a l c u l a t e the long run e l a s t i c i t i e s f o r t h i s s t a t i c expectations model we note that steady state input l e v e l s are given by (obtained by rewriting (4.8) i n terms of reduced form parameters). As we d i d i n Chapter 3 we w i l l take the long run e l a s t i c i t y to mean the r e l a t i v e change i n the target l e v e l s of inputs given a r e l a t i v e change i n input prices or i n output. Since we have found that M i s a stable matrix, by the same reasoning i n Chapter 3 t h i s method of c a l c u l a t i n g long-run e l a s t i c i t i e s gives the same re s u l t s as i f we considered the long run change i n actual values. Consider again the 1975 l e v e l s of c a p i t a l and labour: 18715.3 and 3513.0. x(y, z) = - M {[Ez + G]ye - r t + X} (4.15) 84 Applying equation (4.15) we f i n d the target l e v e l s of those two inputs at 1975 to be 35209.3 and 4384.7; each of these values i s larger than the maximum l e v e l s of inputs observed i n our 1961 to 1984 sample. This i s somewhat su r p r i s i n g , since i n the l i n e a r quadratic model of Chapter 3 we found target l e v e l s (at 1975 at 4 least) much cl o s e r to actual l e v e l s . The long run e l a s t i c i t i e s , c a l c u l a t e d at 1975, are: long run e l a s t i c i t y with respect to: r w y c a p i t a l -0.561 -1.181 +2.327 labour -0.225 -0.831 +1.415 These values are a l l much larger i n absolute terms than those found by Epstein and Denny. Note that the e l a s t i c i t y of c a p i t a l demand with respect to the wage changes sign from the short to the N i c k e l l (1985) points out that the optimal strategy f o r the f i r m i n models of input demand with adjustment costs w i l l not n e c e s s a r i l y involve "asymptotically c l o s i n g the gap between his choice v a r i a b l e and i t s optimal target value...given discounting i t i s not simply worth i n c u r r i n g the a d d i t i o n a l adjustment costs necessary to catch up completely with the [perhaps] growing target" (p. 121). 8 5 long run. This i s due to two f a c t o r s : the large c o e f f i c i e n t ( 1 . 5 2 3 ) for c a p i t a l with respect to lagged labour demand, and the e l a s t i c i t y of labour demand with respect to the wage. While the impact e f f e c t of a change i n the wage i s a movement of c a p i t a l i n the same d i r e c t i o n , there i s a large response of labour i n the opposite d i r e c t i o n to that of the wage change. Af t e r one period t h i s change i n labour demand has a sub s t a n t i a l e f f e c t on c a p i t a l demand, reversing the o r i g i n a l e f f e c t of the change i n the wage on c a p i t a l . The large c o e f f i c i e n t r e l a t i n g the demand for c a p i t a l to lagged labour demand also appeared i n the l i n e a r quadratic model i n Chapter 3, both i n the r e s t r i c t e d and u n r e s t r i c t e d estimation. The e l a s t i c i t y of c a p i t a l with respect to the wage i n that model was p o s i t i v e i n both the short run and the long run. The sign d i d not change because labour was completely i n e l a s t i c with respect to the wage. Compared to the estimates we obtained i n Chapter 3 with the l i n e a r quadratic model, we f i n d with the f l e x i b l e f u n c t i o n a l form a demand f o r c a p i t a l that i s much more e l a s t i c with respect to the re n t a l rate, and a demand for labour that i s , unlike i n Chapter 3, responsive to changes i n the wage and i n the expected d i r e c t i o n . Although i d e n t i c a l data are used to estimate the two models, i t i s not c l e a r exactly what differ e n c e i n the models i s responsible f o r the s u b s t a n t i a l difference i n estimated e l a s t i c i t i e s . A s u r p r i s i n g r e s u l t i s that the e l a s t i c i t y of the demand for c a p i t a l with respect to the wage i s greater than i t s e l a s t i c i t y with respect to the r e n t a l rate. Morrison and Berndt ( 1 9 8 1 , p. 3 5 2 ) also obtain t h i s r e s u l t , a l b e i t i n a model where only c a p i t a l i s treated as a quasi-fixed f a c t o r . 86 The output e l a s t i c i t i e s suggest implausible decreasing returns to scale, casting some doubt on the r e l i a b i l i t y of the other estimated e l a s t i c i t i e s . Epstein and Denny also found output e l a s t i c i t i e s f o r both factors to be greater than one i n the long run, although not to the degree obtained here. As i n the model of Chapter 3, adjustment to target l e v e l s i s slow, so i t i s again i n s t r u c t i v e to ca l c u l a t e medium run e l a s t i c i t i e s . The slow convergence to the long run response with t h i s model i s for the same reasons as i n Chapter 3, that i s that 2 the matrix (I + ( I + M ) + ( I + M ) + ...) i s slow to converge to -M 1 . The f i v e - and ten-year e l a s t i c i t i e s are five-year e l a s t i c i t y with respect to: r w y c a p i t a l -0.139 -0.238 +0.502 labour -0.059 -0.459 +0.696 ten-year e l a s t i c i t y with respect to: r w y c a p i t a l -0.228 -0.437 +0.888 labour -0.094 -0.538 +0.849 We see that f o r c a p i t a l , even a f t e r 10 years neither the e l a s t i c i t i e s with respect to pr i c e s nor the output e l a s t i c i t y have reached one hal f of t h e i r long run values. Even labour demand, which some researchers t r e a t as a variable input, i s remarkably 87 (perhaps implausibly) slow to adjust to i t s long run l e v e l here. The lesson i s that long run e l a s t i c i t i e s must be interpreted with great care, not only i n t h i s study, but i n others as well; f o r example, Epstein and Denny's estimate of the adjustment matrix M i s s i m i l a r to ours, and so we would expect that t h e i r long run e l a s t i c i t i e s are also somewhat misleading. Carmichael, Mohnen, and Vigeant's (1989) estimates of a translog v a r i a b l e cost function with annual Quebec manufacturing data y i e l d a short run e l a s t i c i t y of c a p i t a l with respect to the r e n t a l rate of -0.098, which i s s i m i l a r to our cost function estimate, but t h e i r corresponding long run e l a s t i c i t y i s -0.271, s u b s t a n t i a l l y l e s s than ours (although quite close to Epstein and Denny's U.S. manufacturing estimate). Their short run and long run e l a s t i c i t i e s of labour demand with respect to the wage are -0.118 and -2.354, respectively; t h e i r estimate of the impact e l a s t i c i t y i s smaller than ours but t h e i r long run e l a s t i c i t y i s s u b s t a n t i a l l y l a r g e r . Their short run and long run e l a s t i c i t i e s of c a p i t a l with respect to output are +0.038 and -0.035, and of labour with respect to output are +2.339 and +1.713. Two U.S. studies of cost functions where there are adjustment costs are Pindyck and Rotemberg (1983a) and Kokkelenberg and Bischoff (1986) . Pindyck and Rotemberg, who report only the long run e l a s t i c i t i e s , f i n d c a p i t a l and labour have output e l a s t i c i t i e s of +1.476 and +1.031, and own-price e l a s t i c i t i e s of -2.927 and -0.784, r e s p e c t i v e l y . Kokkelenberg and Bischoff f i n d much lower e l a s t i c i t i e s : c a p i t a l and labour have long run output e l a s t i c i t i e s of + 0.780 and +0.150, and long run own-price e l a s t i c i t i e s of -0.005 and -0.130, r e s p e c t i v e l y . They claim these are "of 88 reasonable magnitude" (p.429). Taking a l l of the above studies together we f i n d , f i r s t , that even over a f a i r l y r e s t r i c t e d c l a s s of models there i s wide disagreement on the magnitudes of e l a s t i c i t i e s , and second, that our estimates of short run and long run own-price e l a s t i c i t i e s f a l l within the range established by previous studies, but that our estimates of input demand e l a s t i c i t y with respect to output are much higher than other studies, e s p e c i a l l y the long run e l a s t i c i t i e s . Epstein and Denny, and Pindyck and Rotemberg, also f i n d evidence of decreasing returns to scale, but not to the same degree as our estimates. 4.3 The Model with Non-Static Expectations Now we change the model somewhat by assuming firms do not expect output l e v e l s and input prices to remain constant, but rather form expectations by a process that we can represent with simple f i r s t order autoregressions. Recall from Chapter 3 that i f the variable to be forecast follows a f i r s t order process, only one observation of the variable, plus estimates of the time series parameters of the vari a b l e , are required to make forecasts. Therefore the reduced form of input demands when expectations are formed t h i s way w i l l not be d i f f e r e n t from the s t a t i c expectations formulation, although the i n t e r p r e t a t i o n of the reduced form w i l l be d i f f e r e n t . Such w i l l be the case here; the reduced form (4.14) w i l l be reestimated, but the s t r u c t u r a l parameters w i l l have d i f f e r e n t estimated values. Correct i d e n t i f i c a t i o n of the s t r u c t u r a l parameters i s important i f p o l i c y analysis using the res u l t s of the estimation i s to be immune from the Lucas c r i t i q u e . 89 We model the evolution of output l e v e l s and input prices as follows: log (yfc) = a + log ( y f c _ 1 ) , (4.16) r f c = u ^ e x p t e ^ ) - D / 9 1 1 + e x p ( 9 1 1 ) r t _ 1 , (4.17) „ t = u 2 ( e x p ( f l 2 2 ) - 1 ) / G 2 2 + exp (e 2 2)w t_ l f (4.18) u where exp(w) denotes e and where the 9 terms are not to be ponfused with t h e i r meaning i n Chapter 3. This i s an unusual method of describing f i r s t order AR processes, but i t allows Epstein and Denny to incorporate the AR parameters into the input demand equations i n a r e l a t i v e l y straightforward way. Equation (4.14) can now be reestimated, but with the reduced form parameters now being defined by M = *[(R - G)*" 1 - I ] , (4.19) E = (R - a ) * * - tf($9 + 9$), (4.20) G = *[(R - a - 9)0 - $u], (4.21) A replaced by R ^ ( R - 9)* - 1A, (4.21) where I i s the i d e n t i t y matrix, 9 i s 2x2 with 6 ^ and 9 2 2 along the diagonal and zeroes o f f the diagonal, a i s 2x2 diagonal with each diagonal element equal to a, /i i s 2x1 c o n s i s t i n g of fx and 90 H^, and A i s 2x1 co n s i s t i n g of parameters and (Epstein and Denny (1983, p. 664)). Note that i f 9 , a, and fi a l l equal 0, i . e . i f expectations are s t a t i c , E, G, M, and X return to t h e i r previous form. Attempts to estimate (4.14) simultaneously with (4.16), (4.17), and (4.18) were not successful, i n p a r t i c u l a r because estimates were always of an unstable M matrix, which means long run e l a s t i c i t i e s would not be well defined, unless the discount rate R was absurdly high. So we followed the method of Epstein and Denny by estimating (4.16), (4.17), and (4.18) separately, and then taking these parameters as given when i t came to re-estimating (4.14), which was estimated with the non-linear maximum l i k e l i h o o d option of SHAZAM.^ Note that care must therefore be taken when considering the estimates of standard errors of the model. It was also necessary to place r e s t r i c t i o n s on * so that we would have a stable M. Since i t was not f e a s i b l e to r e s t r i c t M to be stable i n any general sense i n the estimation, we t r i e d r e s t r i c t i n g ^ to the p a r t i c u l a r values which would give us the reduced form f o r M that we obtained i n the r e s t r i c t e d estimation of the s t a t i c expectations model. Results of the 5 equations, with the r e s t r i c t i o n $ = $ imposed, and with r e s t r i c t e d to This method i s suggested by Wallis (1980) . 91 .4434 -1.2883 .0716 1.0040 are given i n Table 4.2. The l i k e l i h o o d s t a t i s t i c of the 2 r e s t r i c t i o n s i s 3.606, d i s t r i b u t e d % with 5 degrees of freedom, so they are not rejected. The reduced form estimate of M i s the same as f o r the s t a t i c expectations case given above. Other reduced form parameters implied by the Table 4.2 estimates are A " -.1702 +.0011 A .0292 E = , G = +.0029 -.0093 .0756 and the vector which replaces X from the s t a t i c expectations case, as defined by (4.21) i s R 1*(R - 0 ) * 1 A = -3149.991 703.127 The implied values of impact e l a s t i c i t i e s , taken at 1975, are: c a p i t a l labour short run e l a s t i c i t y with respect to: r w y -.044 +.011 +.039 +.004 -.486 +.510 The own-price e l a s t i c i t i e s are barely d i f f e r e n t from the s t a t i c expectations estimates above. Finding the long run e l a s t i c i t i e s i n the non-static expectations case i s somewhat complicated, since we must c a l c u l a t e 92 how the reduced form parameters E and G change when input prices or the l e v e l of output have t h e i r expected future paths permanently s h i f t e d . The long run e l a s t i c i t y with respect to output turns out to be not so d i f f i c u l t to c a l c u l a t e . From equation (4.16) above we see that a permanent proportional increase i n y leaves the underlying exponential growth rate a unchanged. So the reduced form parameters of input demand are unaffected (although input demands are a f f e c t e d by the level change i n output) , and we can r e f e r to equation (4.15), which describes the target input l e v e l s , to f i n d the long run e l a s t i c i t i e s . Regarding long run p r i c e e l a s t i c i t i e s , a long run proportional change i n input p r i c e s would involve a change i n the fx parameters from equations (4.17) and (4.18). This change would a f f e c t the reduced form parameters of the vector G; see equation (4.21). A change i n n changes G by a f a c t o r of Estimates of * and $ are given i n Table 4.2. It turns out for our estimates (and for Epstein and Denny's) that /n and G w i l l move i n the same d i r e c t i o n . This i s s u r p r i s i n g . For estimates where $ and $ have the expected signs (so that own p r i c e e l a s t i c i t i e s are negative and the adjustment matrix M i s stable) we obtain the r e s u l t that adjusting reduced form parameters to take account of a change i n expected future prices a c t u a l l y lowers the e f f e c t s of the p r i c e change. For example, we w i l l see i n simulations below that a permanent f a l l i n r e n t a l rates has a larger p o s i t i v e e f f e c t on input demands i f we do not adjust the reduced form parameters than i f we do make the (correct) adjustment. The converse was the case i n the l i n e a r quadratic model of Chapter 3; the i n t u i t i o n behind 93 the perverse r e s u l t of the model of t h i s chapter i s not obvious. From equation (4.15) and the estimates of reduced form parameters from t h i s section we c a l c u l a t e target l e v e l s of c a p i t a l and labour i n 1975 of 35459.4 and 4416.4. These are nearly i d e n t i c a l to the target l e v e l estimates from the s t a t i c expectations model, and are higher than the actual l e v e l s at that time by factors of about 2 and 1.5 f o r c a p i t a l and labour re s p e c t i v e l y . The long run e l a s t i c i t i e s are: long run e l a s t i c i t y with respect to: r w y c a p i t a l -0.178 -0.971 +2.096 labour -0.055 -0.545 +1.232 Each of these e l a s t i c i t i e s i s smaller i n absolute value than those obtained from the s t a t i c expectations model. Once again, we should look at the medium run e l a s t i c i t i e s , since the adjustment to long run i s very slow. The f i v e - and ten-year e l a s t i c i t i e s are five-year e l a s t i c i t y with respect to: r w y c a p i t a l -0.047 -0.216 +0.459 labour -0.003 -0.248 +0.587 94 ten-year e l a s t i c i t y with respect to: r w y c a p i t a l -0.075 -0.376 +0.805 labour -0.014 -0.311 +0.724 It i s i n t e r e s t i n g to note the estimates of long run e l a s t i c i t i e s i f we do not account f o r the change i n the reduced form parameter G that should occur with a change i n the long run path of p r i c e s . The output e l a s t i c i t i e s are unaffected, because the reduced form parameters of the input demands do not change with a permanent s h i f t of a given proportion to output (although they would change i f the rate of growth of output permanently changed) . With G unadjusted the estimates of long run e l a s t i c i t i e s , obtained from the reduced forms, are long run e l a s t i c i t y (G unadjusted) with respect to: r w y c a p i t a l -0.453 -1.508 +2.104 labour -0.166 -0.993 +1.237 For reasons noted above these are larger i n absolute value than when G i s adjusted to account f o r changed expectations. We also t r i e d estimating the non-static expectations model replacing y i n the input demand equations with y^ ^ and with (1 + a) y - (which could be a proxy f o r expected y ) . There are no c l e a r t h e o r e t i c a l reasons for choosing one s p e c i f i c a t i o n over another here. The logs of the l i k e l i h o o d functions from using y , r e s p e c t i v e l y . There are no changes i n the signs of the s t r u c t u r a l parameters. F i n a l l y , we attempted estimating Epstein and Denny's model imposing the short run e l a s t i c i t i e s we obtained i n the l i n e a r quadratic model of Chapter 3 . We are i n t e r e s t e d i n whether t h i s r e s t r i c t i o n s u b s t a n t i a l l y lowers the goodness of f i t of the model. In the model of t h i s chapter, own p r i c e e l a s t i c i t i e s are determined by our estimate of E, which i n turn i s determined by the s t r u c t u r a l parameter $. We seek only to r e s t r i c t the short run e l a s t i c i t y (we cannot simultaneously r e s t r i c t the long run e l a s t i c i t y , since that would e n t a i l a complicated r e s t r i c t i o n on the adjustment process). R e c a l l that i n Chapter 3 we obtained e l a s t i c i t y of c a p i t a l demand with respect to the r e n t a l rate and the wage of - . 0 0 4 and + . 0 0 4 respectively, and a labour demand i n e l a s t i c with respect to both the r e n t a l rate and the wage. These are the short run e l a s t i c i t i e s we w i l l impose here. Note that i n Chapter 3 the r e n t a l rate and wage were defl a t e d by the output p r i c e index, rather than the materials p r i c e index, so the comparison i s not p r e c i s e . Given our imposed values of R, a, and 6, we impose a value of $ of t and (1 - a)y t - 1 are 2 3 6 . 3 5 9 , 2 1 7 . 6 9 4 , and 2 1 9 . 5 5 6 , - . 0 6 1 8 + . 0 0 2 8 f - . 0 0 7 2 + . 0 0 3 3 i n order to obtain a value f o r E of 96 -.0157 +.0004 0 0 and the e l a s t i c i t i e s given above. When the system i s estimated, the only free parameters are <f>, and A; f i v e parameters i n a l l . There are eight r e s t r i c t e d parameters. The u n r e s t r i c t e d l og of the l i k e l i h o o d function i s 238.16, and the r e s t r i c t e d r e s u l t i s 188.84, so t h i s r e s t r i c t i o n i s c l e a r l y rejected. Since so many aspects of the l i n e a r quadratic model r e s u l t s were unsatisfactory, we are not altogether surprised that the e l a s t i c i t i e s obtained with that model are rejected by the model of t h i s chapter. This model with non-static expectations w i l l now be used f o r simulations comparable to those done i n Chapter 3 with the al t e r n a t i v e model. 4.4 Simulations with Non-Static Expectations Table 4.3 records the re s u l t s of simulations which are analogous to those i n Table 3.5 from the l i n e a r quadratic model. Column A l i s t s actual data f o r c a p i t a l i n the manufacturing sector. Column B gives a long term forecast one might have made i n 1975 using the f l e x i b l e f unctional form model with non-static expectations as estimated above. At 1975, output i n 1984 i s forecast at 88170.8 while the actual value was 76700.9, the re n t a l rate i s forecast at .128 while the actual values were .130 i n 1983 and .146 i n 1984, and the wage i s forecast at 4.43 while the actual value was 4.65. Labour i s forecast to be 4062.1 i n 1984 while the actual value was 3621.0. We see i n the table that the 97 c a p i t a l stock throughout the 1980s would have been s l i g h t l y over-estimated with t h i s model. In simulation C there i s a 10% negative shock to the r e n t a l rate i n 1975, but i t i s temporary i n that the underlying parameters of the time seri e s process of the r e n t a l rate, f i ^ and 6^, are l e f t unchanged. By 1984 the e f f e c t s of the shock on the r e n t a l rate i t s e l f has almost completely d i s s i p a t e d : the 1984 base case (column B) forecast r e n t a l rate i s .12775, and the one-off shock case (column C) forecast r e n t a l rate i s .12621. But c a p i t a l and labour demand depend heavily on t h e i r previous values, and even at 1984 there are s i g n i f i c a n t l i n g e r i n g e f f e c t s from the 1975 shock. C a p i t a l i n 1984 i s 1.10% higher than i t otherwise would have been and labour i s 4074.6, which i s 0.31% higher than i t otherwise would have been. R e c a l l that the i d e n t i c a l shock i n the l i n e a r quadratic model l e f t 1984 c a p i t a l only 0.02% higher than the base case, and labour was completely unaffected by the shock. In columns D and E are l i s t e d the forecast paths of the c a p i t a l stock when i n 1975 the r e n t a l rate i s permanently lowered by 10% from i t s base case path. In column E the reduced form parameters have been a l t e r e d as they should be given the change i n regime, while i n column D the base case reduced form i s used. The path of r e n t a l rates i s lowered permanently here by c u t t i n g the 1975 value of r by 10%, from .1106 to .0950, and c u t t i n g n by 10%, from .02819 to .02537. The reduced form of the r e n t a l rate path i s changed from r = .0254 + .8071r t_ 1 98 to r = .0228 + .7069r , . t t-1 As explained above the change i n changes the vector G, which i s part of the reduced form of the input demand equation. G changes from ,0292 .0756 to .0283 .0756 From equation (4.15), which describes the target l e v e l s of inputs, we see that the target l e v e l s x depend p o s i t i v e l y on G ( a l l of the elements of -M ^ are positive) , so f o r given current l e v e l s of input prices the permanent reduction i n future r e n t a l rates a c t u a l l y lowers the input targets. The r e s u l t surprises because estimated own p r i c e e l a s t i c i t i e s are negative. The change i n the reduced form parameters has s i g n i f i c a n t e f f e c t s . Comparing the forecasts f o r 1984 l e v e l s , the c a p i t a l stock f o r the case where reduced form parameters of the input demand equations are adjusted i s 1.72% higher than the base case forecast, while when these parameters are not adjusted i t i s 3.12% higher than the base case. The diffe r e n c e i s the same on the labour side; the column E (adjustment made) 1984 forecast i s 4077.6, 0.38% higher than the base case, while the column D (adjustment not made) forecast i s 4092.2, 0.74% higher than the base case. 4.5 Comparing the Performance of the Models Both the l i n e a r quadratic r a t i o n a l expectations model of Chapter 3 and the f l e x i b l e f u n c t i o n a l form model of t h i s chapter seek to explain the manufacturing sector's demand for c a p i t a l and labour. The models are non-nested; r e c a l l that i n the l i n e a r quadratic model input prices were de f l a t e d with an output p r i c e index, whereas with the f l e x i b l e f u n c t i o n a l form model the parameters of the value function of a r e s t r i c t e d v a r i a b l e cost function required d e f l a t i n g q u a s i - f i x e d input p r i c e s with a vari a b l e input p r i c e index. Also, the rate of output i s assumed to be determined exogenously i n the l a t t e r , and appears as an explanatory variable i n the input demand equations, while i n the former output i s endogenous and does not play a part i n the estimation. We now examine the r e l a t i v e performance of the two models with a t e s t proposed by Davidson and MacKinnon (1981) ; s p e c i f i c a l l y t h e i r "P t e s t " . Using t h e i r notation, l e t the input demand equations of the l i n e a r quadratic model of Chapter 3 be written i n the form x t = f (X t, |3) + e Q t , (4.22) and the input demand equations of the f l e x i b l e f u n c t i o n a l form model be written i n the form 100 x t = g ( z t . K) + e l t, (4.23) where x i s the vector of input demands ( c a p i t a l and labour), X and Zfc are the data used to estimate the two models, and |3 and y are the parameters to be estimated. Following Davidson and MacKinnon, define A A f f c = f ( X t , /3), (4.24) and <3T = g(Z t, V), (4.25) A A where /3 and y are the maximum l i k e l i h o o d estimates of the r e s t r i c t e d dynamic expectations models, given i n Tables 3.4 and 4.2 res p e c t i v e l y . We te s t the n u l l hypothesis (4.22) against the a l t e r n a t i v e hypothesis (4.23) by estimating the regression xfc - f t = a ( g t - f t ) + x tb + e f c. (4.26) where b i s a vector of unknown parameters and i s an error term.If the n u l l hypothesis i s true,.the true value of a i s zero. If the a l t e r n a t i v e hypothesis i s true, the estimate of a should converge asymptotically to one. The estimate of a i n (4.26) does not have an asymptotically v a l i d standard er r o r unless we also include on the right-hand side a vector representing the d e r i v a t i v e s at each observation of the A predicted values of the model with respect to i t s parameters: F i n Davidson and MacKinnon's notation. If the model f (X , /3) i s A t l i n e a r , then F i s X^_. The l i n e a r quadratic model i s l i n e a r , although with non-linear r e s t r i c t i o n s , so we include X̂ _ i n (4.26) . The elements of X are the righ t hand side variables of the l i n e a r quadratic model of Chapter 3: k , 1 , r. ,, w , and a constant (note that r and w are d e f l a t e d with an output p r i c e A A index i n Chapter 3). The ser i e s f and g were generated from the estimates given i n Tables 3.4 and 4.2. Equation (4.2 6) was estimated separately for c a p i t a l and labour, by ordinary l e a s t squares. For c a p i t a l the estimate of a is. 0.807, with a t - s t a t i s t i c of 1.419 (there are 17 degrees of freedom), and for the labour the estimate of a i s 1.095, with a t - s t a t i s t i c of 9.934. This i s strong evidence that we should r e j e c t the model (4.22), the l i n e a r quadratic model. This does not mean we should n e c e s s a r i l y accept the a l t e r n a t i v e model. To t e s t the model of t h i s chapter, the f l e x i b l e f u n c t i o n a l form, we would have to construct the t e s t so that the f l e x i b l e f unctional form was the n u l l hypothesis. We do the following regression: x t " gt = a ( ft " gt) + Gtd + V (4.27) A where v i s an error term, d i s a vector of parameters, and Gfc represents the d e r i v a t i v e s of the predicted values of y) 102 with respect to the parameters, namely r^y^(1.01911) t , w ty t(1.01911) t , y (1.01911) _ t, and a constant, i n both the c a p i t a l and labour equations, where 1.01911 i s the estimate of y from Table 4.2. Input p r i c e s r^ and ŵ  are here d e f l a t e d by the materials input p r i c e index. Also included i n the c a p i t a l equation i s - t (1.01911) ( - t - 1 ) ( E 1 1 r t y t + E^w y + G^y ) f and i n the labour equation - t (1.01911) ( - t _ 1 ) ( E ^ r ^ + E ^ w ^ + G 2y ) , where these are the d e r i v a t i v e s of the predicted values of the f l e x i b l e f u n c t i o n a l form model with respect to y, and where E^_. and G^, i = 1, 2 are from the reduced form estimates given above. For c a p i t a l the estimate of a i s 2.545, with a t - s t a t i s t i c of 0.2 97 (there are 15 degrees of freedom), and for the labour the estimate of a i s -790. 97, with a t - s t a t i s t i c of -1.408. The 5% c r i t i c a l value of the t - s t a t i s t i c i s 1.753, so we f a i l to rej e c t the f l e x i b l e f u n c t i o n a l form i n the presence of the l i n e a r quadratic model as an a l t e r n a t i v e hypothesis. F i n a l l y , we ask whether what appears to be superior performance by Epstein and Denny's model r e l a t i v e to the l i n e a r quadratic model i s due s o l e l y to the i n c l u s i o n of output as an explanatory variable i n the former but not the l a t t e r . We begin by regressing ĝ _, the predicted values from the model of t h i s chapter, on output and a constant. The r e s u l t s are, for c a p i t a l gk = -2825.4 + -344y , (4.28) and f o r labour g l f c = 2475.5 + .018y t. (4.29) 103 We t r y Davidson and MacKinnon's P t e s t of the l i n e a r quadratic model using the predicted values of the above equations as the a l t e r n a t i v e hypothesis, rather than g^. The r e s u l t s are that we s t i l l r e j e c t the n u l l hypothesis of the l i n e a r quadratic model, even when set against simply that part of the predicted values of Epstein and Denny's model explained by current output. The estimate of a for the c a p i t a l equation i s .113 with a t - s t a t i s t i c of 2.346, and f o r labour a i s 2.219, with a t - s t a t i s t i c of 7.954. This suggests that the i n c l u s i o n of output i n one model but not the other explains at le a s t to some degree the d i f f e r e n t l e v e l s of performance of the models. It i s also the case, however, that the l i n e a r quadratic model i s rejected by the Epstein and Denny model even when we discount f o r the e f f e c t s of the i n c l u s i o n of output as an explanatory v a r i a b l e . The estimated residuals of equations (4.28) and (4.29) could be thought of as the part of the demands for c a p i t a l and labour explained by the "non-output v a r i a b l e s " of the Epstein and Denny model. We performed regression (4.2 6) f o r c a p i t a l and labour using the residuals from (4.28) and (4.29) i n place of g . The re s u l t s were an estimate f o r a of .114 i n the c a p i t a l equation, with a t - s t a t i s t i c of 2.461, and .759 i n the labour equation, with a t - s t a t i s t i c of 10.270. Again we r e j e c t the l i n e a r quadratic model. 4.6 Conclusions The dynamic input demand model of Epstein and Denny (1983) has been estimated f or both s t a t i c expectations and for r a t i o n a l 104 expectations, where by r a t i o n a l expectations we mean that the parameters describing the time series processes of input prices and output were incorporated into the input demand equations. In general the model performed well; a symmetry r e s t r i c t i o n was s a t i s f i e d , and parameter estimates were of the expected sign. E l a s t i c i t i e s were i n the (large) range of estimates of other researchers, although estimates of output e l a s t i c i t i e s were implausibly high. In addition, according to the r e s u l t s of non-nested hypothesis t e s t s between the model of t h i s chapter and the l i n e a r quadratic model of Chapter 3, the l i n e a r quadratic model i s rejected, but the model of t h i s chapter i s not. We found a paradoxical r e s u l t when i t came to noting how target l e v e l s of input demand changed when the time series path of input p r i c e s changed. While with the l i n e a r quadratic model of Chapter 3 we found that ignoring the e f f e c t of the path of input prices on the reduced form parameters of input demand would lead one to underestimate c a p i t a l ' s own p r i c e e l a s t i c i t y , i n the model of t h i s chapter ignoring t h i s e f f e c t would cause one to overestimate the e l a s t i c i t y . The r e s u l t s of the models of Chapters 3 and 4 are so d i f f e r e n t with regards to estimated e l a s t i c i t i e s and to the changes i n long run e l a s t i c i t i e s when we change the s p e c i f i c a t i o n of expectation formation, that i n the end we know l i t t l e about the s i z e or even the sign of the e f f e c t s of heeding the Lucas c r i t i q u e i n dynamic models of f a c t o r demand. 105 TABLE 4.1 R e s t r i c t e d Estimates of F l e x i b l e Functional Form Model with S t a t i c Expectations Parameter Estimates 11 21 *1 11 22 .19120 (.03325)' -.04199 (.02292) -2890.4 (560.18) 18.372 (11.877) -25.405 (15.504) -.19199 (.10463) sigma .11811x10 -4 .13303x10 12 22 12 .55536x10 -1.5230 (.34998) .97260 (.07449) 319.82 (451.98) 2.1133 (1.1012) -1.3022 (1.2419) .02566 (.00719) -6 log l i k e l i h o o d function = 234.4712 u n r e s t r i c t e d 0 log l i k e l i h o o d function = 238.1624 marginal s i g n i f i c a n c e of r e s t r i c t i o n = .007 a standard errors of parameter estimates i n parentheses. b sigma i s the variance-covariance matrix of the 2 equation system where the order of the equations i s , by dependent variable, k/y, 1/y. c u n r e s t r i c t e d estimate does not impose $ 0 = $ . 106 TABLE 4.2 R e s t r i c t e d Estimates of F l e x i b l e Functional Form Model with Autoregressive Expectations. Exogenous Parameters H 1 .04456 (.01842) ^ 2 .32748 (.18897) 911 - - 3 4 6 9 ° (.14197) 0 -.06908 (.05060) a .04146 (.01018) Technology Parameters A -856.05 (73.123) <P 1.2184 (.25150) -1.0356 (.23971) -.06883 (.00835) 11 »22 sigma ,11497x10 -4 ,15097x10 -5 12 235.89 (76.501) .76094 (.06895) -.09908 (.03174) .01911 (.00249) .55177x10 log l i k e l i h o o d function = 236.3593 un r e s t r i c t e d l og l i k e l i h o o d function = 238.1624 marginal s i g n i f i c a n c e of r e s t r i c t i o n = .607 a standard errors of parameter estimates i n parentheses. b sigma i s the variance-covariance matrix of the 2 equation system where the order of the equations i s , by dependent variable, k/y, l / y . c u n r e s t r i c t e d estimate does not impose $ 1 2 = o r t h e v a l u e s f o r $ used to ensure a stable adjustment matrix. 107 TABLE 4.3 Forecast values of the c a p i t a l stock under various conditions, using the f l e x i b l e f u n c t i o n a l form model. Simulation Year A B C D E 1975 18715 .3 18862 .2 18945 .1 18945 .1 18903 .6 1976 19404 .4 19611 .9 19750 .9 19771 .2 19692 .4 1977 20004 .6 20325 .8 20507 .6 20562 .5 20447 .2 1978 20376 .6 21018 .1 21232 .2 21332 .9 21182 .0 1979 21013 .3 21696 .4 21933 .9 22089 .4 21903 .7 1980 21982 .7 22366 .7 22620 .9 22838 .0 22618 .2 1981 23286 .0 23034 .5 23299 .7 23584 .0 23330 .7 1982 23805 .9 23704 .5 23976 .1 24331 .7 24045 .6 1983 23698 .8 24380 .8 24655 .3 25085 .3 24766 .8 1984 23603 .4 25066 .8 25341 .4 25848 .2 25497 .9 Description of Simulations A: Actual data f o r c a p i t a l stock (machinery and equipment). B: Simulated forecast using f l e x i b l e f u n c t i o n a l form model, with parameter estimates from Table 4.2, s t a r t i n g at 1975. C: Simulated forecast, using the model of B, with a one-off negative shock to the r e a l r e n t a l rate of c a p i t a l i n 1975 of 10%. D: Simulated forecast, using the model of B, with a permanent lowering of the path of the r e a l r e n t a l rate by 10%, beginning i n 1975, where firms do not r e a l i z e there has been a change i n regime. E: Simulated forecast, with a permanent lowering of the path of the r e a l r e n t a l rate by 10%, beginning i n 1975, where the reduced form parameters of the model have been adjusted to r e f l e c t the change i n the path of r e n t a l rates ( i . e . where firms do r e a l i z e there has been a change i n regime). 108 CHAPTER 5 Conclusion In t h i s t h e s i s we have used data from Canadian manufacturing from 1961 to 1984 to estimate two d i s t i n c t models of dynamic input demand, each of which to some degree incorporates the hypothesis of r a t i o n a l expectations. The a l t e r n a t i v e models generated quite d i f f e r e n t estimates of the e l a s t i c i t i e s of input demands with respect to t h e i r p r i c e s . Because of t h i s , and because of other uncertainties about the most appropriate ways of modelling expectations about tax p o l i c y and other components of the r e n t a l cost of c a p i t a l , no d e f i n i t i v e conclusions could be reached on the e f f e c t s of the r e n t a l rate of c a p i t a l on investment. There i s wide disagreement, or at least uncertainty, about the importance of the r e n t a l cost of c a p i t a l on business investment. We have here attempted to obtain r e s u l t s on t h i s question using recently derived techniques for estimating dynamic models, but we have only succeeded i n increasing our confusion on the question. Estimates of a l i n e a r quadratic r a t i o n a l expectations model, as s p e c i f i e d i n Epstein and Yatchew (1985), were of a demand for c a p i t a l quite i n e l a s t i c with respect to the r e n t a l rate. Estimates of a dynamic model using a f l e x i b l e f u n c t i o n a l form, as derived by Epstein and Denny (1983), were of a c a p i t a l demand with much greater e l a s t i c i t y with respect to the r e n t a l rate i n the long run. When the models were compared using the P t e s t of Davidson and MacKinnon (1981), Epstein and Denny's model seemed to outperform the l i n e a r quadratic model, although we 109 were g i v e n some r e a s o n t o b e l i e v e t h a t t h i s m i ght be due t o t h e i n c l u s i o n o f o u t p u t as an e x p l a n a t o r y v a r i a b l e i n t h e f o r m e r b u t not i n t h e l a t t e r . When t h e s e two models were u s e d t o e s t i m a t e t h e e f f e c t s o f a change i n t h e t i m e s e r i e s p a t h o f t h e r e n t a l r a t e o f c a p i t a l , where t h e s i m u l a t i o n s were done i n such a way as t o a v o i d t h e Lucas c r i t i q u e o f e c o n o m e t r i c p o l i c y e v a l u a t i o n , t h e f l e x i b l e f u n c t i o n a l f o r m model's p r e d i c t i o n was o f a much l a r g e r change t o t h e medium and l o n g t e r m c a p i t a l s t o c k t h a n t h e change p r e d i c t e d by t h e l i n e a r q u a d r a t i c model. Each o f t h e s e e s t i m a t e s i n t u r n was q u i t e d i f f e r e n t f r o m t h e p r e d i c t i o n s o f t h e c o r r e s p o n d i n g e l a s t i c i t i e s f r o m e s t i m a t e s o f t h e two models w i t h s t a t i c e x p e c t a t i o n s . E s t i m a t e s o f t h e e f f e c t s o f a change i n t h e p a t h o f t h e r e n t a l r a t e where we i g n o r e d t h e .warning o f t h e Lucas c r i t i q u e gave an even wider range o f e s t i m a t e s . We f o u n d t h a t whether we a d j u s t e d i n p u t demand r u l e s f o r v a r i o u s r e n t a l r a t e regimes (as Lucas w o u l d s u g g e s t we s h o u l d do) made a l a r g e d i f f e r e n c e t o o u r r e s u l t s , b u t t h e d i r e c t i o n o f t h e b i a s f r o m not making t h e a d j u s t m e n t s was ambiguous. These r e s u l t s s u g g e s t t h a t we s t i l l know v e r y l i t t l e about i n p u t demands when t h e r e a r e a d j u s t m e n t c o s t s , and t h a t t h e r e i s much more t o be done i n t h i s a r e a . L i s t e d below a r e some o f t h e many q u e s t i o n s t h a t r e m a i n . What a r e t h e e f f e c t s on o u r r e s u l t s o f u s i n g a g g r e g a t e d a t a t o e s t i m a t e a model o f a " r e p r e s e n t a t i v e f i r m " ? The model we e s t i m a t e i n C h a p t e r 4 i s a t l e a s t p o t e n t i a l l y c o n s i s t e n t w i t h t h e 110 aggregation of a number of firms, according to conditions f o r consistent aggregation set out i n Blackorby and Schworm (1982), although we d i d not t e s t whether these conditions were s a t i s f i e d . Whether the l i n e a r quadratic model estimated i n Chapter 3 i s consistent with respect to aggregation i s less c l e a r . Geweke (1985) suggests that the biases from aggregation i n such models might e a s i l y be as large as any biases from ignoring the warnings of the Lucas c r i t i q u e . It would be of i n t e r e s t to f i n d whether t h i s i s true with actual data. Can more i n t e r e s t i n g s p e c i f i c a t i o n s of technology be developed? We mentioned above that a dynamic model of the f i r m with r a t i o n a l expectations, a f l e x i b l e f u n c t i o n a l form, and endogenous output has never been estimated. Bernstein and N a d i r i (1987) suggest that i t would also be of i n t e r e s t to incorporate va r i a b l e u t i l i z a t i o n rates i n dynamic models, which could have the e f f e c t of endogenizing the depreciation rate of c a p i t a l . F i n a l l y , although t h i s thesis has been purely a p o s i t i v e analysis of manufacturing i n Canada and r e n t a l rates, we could i n future attempt to use improved estimates of the p r i v a t e sector's response to tax p o l i c y to help answer some normative questions. Woodward (1974) and Kesselman, Williamson, and Berndt (1977) have questioned whether tax incentives f o r investment are the most appropriate device f o r dealing with unemployment problems. 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