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UBC Theses and Dissertations

An approach toward a new synthesis of urban economics and urban geography Redfern, Paul Anthony 1983

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AN APPROACH TOWARD A NEW SYNTHESIS OF URBAN ECONOMICS AND URBAN GEOGRAPHY By PAUL ANTHONY REDFERN B.A., St Catherine's College, Oxford Un i v e r s i t y , England, 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS In THE FACULTY OF GRADUATE STUDIES (Department of Geography) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1 9 8 3 (3)Paul Anthony Redfern 1 9 8 3 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. requirements for an advanced degree at the University Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 3A ^ %b 3E-6 (.3/81) ABSTRACT C o n v e n t i o n a l a c c o u n t s o f t h e r e l a t i o n between employment and r e s i d e n t i a l l o c a t i o n i g n o r e t h e g r o w i n g e v i d e n c e o f sometimes s e r i o u s m i smatches between t h e two p a t t e r n s . T h i s t h e s i s d i s c u s -s e s t h e models u s e d i n t h e s e a c c o u n t s i n t h e c o n t e x t o f t h e d e b a t e i n e c o n o m i c s o v e r c a p i t a l t h e o r y , s p e c i f i c a l l y t h e c o n s e -quences f o r l o c a t i o n a l a n a l y s i s o f a s s u m i n g t h a t a l l i n v e s t m e n t s a r e i n s t a n t a n e o u s l y and c o s t l e s s l y t r a n s f o r m a b l e . Economic p r e s -s u r e s would t h e n c a u s e employment and r e s i d e n t i a l l o c a t i o n p a t -t e r n s t o be complementary a t a l l t i m e s , r e g a r d l e s s o f whether t h e v e r s i o n u s e d was u t i l i t y t h e o r y , C obb-Douglas, c o n s t a n t o r v a r i -a b l e e l a s t i c i t y o f s u b s t i t u t i o n , o r p r o d u c t i o n f u n c t i o n v i n t a g e a p p r o a c h e s . The r e a s o n s f o r making s u c h a s s u m p t i o n s a r e t o be f o u n d i n t h e h i s t o r y o f t h e d e b a t e between c l a s s i c a l and n eo-c l a s s i c a l v i e w s on t h e o b j e c t i v e s o f economic a n a l y s i s . Neo-R i c a r d i a n e c o n o m i c s , l e a d i n g t h e c l a s s i c a l r e v i v a l , has been a p p l i e d t o t h e u r b a n s i t u a t i o n , b u t w i t h l i t t l e s u c c e s s . C o n s i s -t e n t on i t s own t e r m s , i t i s s i m p l y n o t d e s i g n e d f o r t h e a n a l y s i s o f a s pace economy, c h a r a c t e r i z e d by t h e e x i s t e n c e o f t r a n s p o r t c o s t s and economies o f s c a l e . The a t t e m p t , however, does r e v e a l p r e v i o u s l y u n s u s p e c t e d i n c o n s i s t e n c i e s i n A l o n s o ' s t h e o r y o f u r b a n l o c a t i o n . A n o t h e r a l t e r n a t i v e i s p o s t - K e y n e s i a n t h e o r y , w h i c h r e -g a r d s K e y n e s 1 work as c o m p l e t i n g t h e s t r u c t u r e o f c l a s s i c a l e c o n o m i c s r a t h e r t h a n as a l i m i t i n g c o n d i t i o n w i t h i n t h e f r a m e -work o f n e o - c l a s s i c a l e c o n o mics ( t h e n e o - K e y n e s i a n p o s i t i o n ) . T h i s p r o v i d e s t h e b a s i s f o r a p o w e r f u l c r i t i q u e o f g e n e r a l e q u i -l i b r i u m t h e o r y , on t h e b a s i s o f w h i c h i t i s p o s s i b l e t o r e f u t e t h e arguments a g a i n s t r e n t c o n t r o l . The r e m a i n d e r o f t h e t h e s i s i s c o n c e r n e d w i t h e l a b o r a t i n g t h e c o n s e q u e n c e s o f a d e m o n s t r a t i o n t h a t t h e c o n c e p t o f e f f e c t i v e demand, p r e v i o u s l y t h o u g h t t o be w h o l l y m acro-economic i n c h a r a c t e r , may i n f a c t be d i s a g g r e g a t e d . A model o f an u r b a n economy i n w h i c h a h o u s i n g s h o r t a g e may a r i s e and p e r s i s t c a n t h e r e f o r e be d e v e l o p e d . E f f e c t i v e demand i n any s e c t o r , c l a s s o r r e g i o n depends on t h e amount o f i n v e s t m e n t c a r r i e d o u t i n t h a t s e c t o r , c l a s s o r r e g i o n . An i n d i v i d u a l ' s e f f e c t i v e demand f o r p r i v a t e l y owned h o u s i n g depends on t h e amount mortgage conmpanies a r e p r e p a r e d t o i n v e s t i n t h a t i n d i v i -d u a l ' s p r o s p e c t s ( o r has i n v e s t e d i n t h e p a s t i n t h e c a s e o f t r a d i n g u p ) . The s t a t e o f c o m p e t i t i o n f o r i n v e s t m e n t f i n a n c e o v e r a l l d e t e r m i n e s t h e t o t a l amount mortgage companies have a v a i -l a b l e t o i n v e s t . G e n e r a l l y , t h e s t a t e o f c o m p e t i t i o n makes i t l i k e l y t h a t t h e amount a v a i l a b l e w i l l be i n s u f f i c i e n t . I n s u c h a s i t u a t i o n , m i s m a t c h e s between employment and r e s i d e n t i a l l o c a t i o n w i l l a r i s e . However, as a l l l o c a t i o n d e c i s i o n s i n v o l v e i n v e s t m e n t o u t l a y s , and c o n t r o l o v e r i n v e s t m e n t i m p l i e s s o c i a l c o n t r o l , t h i s model has f u r t h e r c o n s e q u e n c e s f o r l o c a t i o n and s o c i a l t h e o r y . I n p a r t i c u l a r , an a c c o u n t i s g i v e n o f g e n t r i f i c a t i o n w h i c h depends o n l y on c h a n g i n g i n v e s t m e n t o p p o r t u n i t i e s o v e r t i m e and n o t on any d u b i o u s a s s e r t i o n s a b o u t new c l a s s e s . I n f a c t , t h i s a n a l y s i s p r o v i d e s a c o n t e x t f o r s o c i o l o g i c a l t h e o r i e s o f u r b a n d i f f e r e n -t i a t i o n , i n p a r t i c u l a r u r b a n m a n a g e r i a l i s m and h o u s i n g c l a s s e s . T h e s e depend on a W e b e r i a n s y s t e m o f s o c i a l c l a s s i f i c a t i o n , b u t t h e K e y n e s i a n p e r s p e c t i v e a d o p t e d h e r e d e m o n s t r a t e s how t h i s may be linked to the c l a s s i c a l and Marxist The intimate connection between forged by the process of investment as therefore advanced as a new means of urban economics and s o c i a l geography. categories of class, location and s o c i a l class indicated by this model i s achieving a synthesis of i v ACKNOWLEDGEMENTS I would l i k e t o t a k e t h i s o p p o r t u n i t y t o t h a n k a l l t h o s e who have h e l p e d t o make my s t u d i e s a t U.B.C. t h e most i n s t r u c -t i v e , and a l s o some o f t h e h a p p i e s t , y e a r s o f my l i f e . W i t h o u t t h e i r s u p p o r t , c a r e and g e n e r o s i t y I s h o u l d n e v e r have been a b l e t o g e t t h i s f a r . I n p a r t i c u l a r , I would l i k e t o m e n t i o n J o h n Radke, Ben M o f f a t and S c o t t Graham, P e t e r Bowra, I a n S p i e r s , Mai Summersby, Mai and Sue J o n e s , M e r s e y s i d e r s F.C., J o a n i e and Tommy M a c N e i l l , L a u r i e and L e s M i l d e n e r , CUPE 2278, S t u a r t P a r k i n s , Kim C o n o l l y and S h i e l a Dodd. I would a l s o l i k e t o t h a n k D r s . C e r i P e a c h and J o hn M e r c e r f o r , r e s p e c t i v e l y , s u g g e s t i n g I a p p l y and i n v i t i n g me t o come t o U.B.C. My t h a n k s a l s o t o Dr. A.L. F a r l e y f o r whose s i n c e r i t y , encouragement and i n t e r e s t I d e e p l y a p p r e -c i a t e . Above a l l , however, I would l i k e t o t h a n k t h e f o l l o w i n g : my s u p e r v i s o r , D r . Ken D e n i k e , who has b o r n e t h e b u r d e n o f h a v i n g me as one o f h i s s t u d e n t s w i t h a f a r g r e a t e r sympathy and g e n e r -o s i t y t h a n I had any r i g h t t o e x p e c t ; and my s e c o n d r e a d e r , Dr. D e n n i s C a p o z z a , who a l s o gave g e n e r o u s l y o f h i s t i m e and e x p e r -t i s e , r e g a r d l e s s o f o u r d i f f e r e n c e s o v e r economic p h i l o s o p h y . As f o r Raymon T o r c h i n s k y and C h r i s t i n e Gordon, words c a n n o t e x p r e s s t h e d e b t I owe them. My p a r e n t s , K a t e and B i l l , a l o n g w i t h my b r o t h e r s L i a m , D a v i d and J a c k , have g i v e n me e v e r y t h i n g anyone c o u l d w i s h f o r ; i n r e t u r n , I s h o u l d l i k e t o d e d i c a t e t h i s t h e s i s t o my f a m i l y , and t o Raymon and C h r i s t i n e . TABLE OF CONTENTS Introduction 1 1 Neo-Classical Approaches To Urban Processes: Review Of Methodology 26 1.1 U t i l i t y Theory 27 1.2 Production Theory 38 1.3 Technical Change 52 2 Neo-Classical Theory In Urban Economics I: U t i l i t y Theory Approaches 58 3 Neo-Classical Theory In Urban Economics II: Non—Vintage Production Function Approaches 65 3.1 Cobb-Douglas Approaches 69 3.2 Constant E l a s t i c i t y Of Substitution 73 3.3 Variable E l a s t i c i t y Of Substitution 83 3.4 Non—Vintage Production Function Models: Conclusion . 89 4 Production Function Vintage Approaches 91 5 Critiques Of Neo-Classical Analysis: The Neo—Ricardian Alternative 104 5.1 Introduction . .. 104 5.2 C l a s s i c a l Economics 109 5.3 Neo-Classical Economics And Marginal Theories Of D i s t r i b u t i o n : Discussion And Critique 111 5.4 The Sraffa System 119 5.5 Samuelson vs. Garegnani 131 5.6 Nuti On Vintage Models And The Value Of Capital 139 5.7 Sraffan And Marxist Theories Of Rent 144 v i 6 Neo—Ricardian Approaches To Urban Analysis 152 6.1 Von Thunen — Based Land-Use Models 152 6.2 Reswitching P o s s i b i l i t i e s In Neo—Ricardian Land-Use Models 168 6 . 3 Conclusion 181 7 Post—Keynesian Approaches To Urban Growth 184 7.1 Keynes, Kalecki And The Foundations Of Post—Keynesian Economics 187 7.2 E f f e c t i v e Demand And Location Theory I 193 7.3 Characteristics Of Investment I 196 7.4 Characteristics Of Investment II 202 7.5 Patterns Of Long Run Urban Growth 209 7.6 Autonomous And Induced Investment 235 8 Post—Keynesian Approaches To Urban Analysis 238 8.1 Space And Economic Theory 238 8.2 The Orthodox Theory Of Housing Supply And Demand ...241 8.3 The Keynesian Critique Of Micro—Economic Price Theory 255 8.4 E f f e c t i v e Demand And The Provision Of Housing: The Short Run 268 8.5 E f f e c t i v e Demand And The Provision Of Housing: The Long Run 284 8.6 Housing Supply In A Keynesian Perspective 294 8.7 E f f e c t i v e Demand And Location Theory II 304 8.8 Conclusion 323 9 Implications For Urban Social Theory And Analysis 327 v i i Appendix A: Note On Symbols 346 Bibliography 348 v i i i L i s t of Figures Number Page T i t l e 1.1 8 Trade-off Possibilities Under Opposing Assumptions On I n d i v i s i b i l i t i e s 1.1 33 Income Consumption Paths, Or Engel Curves 1.2 49 Isoquants Of The VES Production Function 1.3 54 Hicks- and Harrod-neutral Technical Progress 5.1 124 Pair Of Techniques Exhibiting Differing Wage-Interest Relationships 5.2 134 ( Factor-Price Frontier Of Techniques With Constant Q/L Ratios, Ordered by K/L Ratios 6.1 156 Land Rents/Unit Area And Land Use On An Isotropic Plain 6.2 163 Land Rents, Prices and Transport Costs On An Isotropic Plain 6.3 170 Behaviour Of Wage-Rent Frontier For Changing Ratios Of W/w At Different Levels Of The Interest Rate 6.4 171 Rent-Profit Relation In A Two Sector Economy 6.5 171 Wage-Profit Relation In The Same Economy 6.6 171 Composite Graph Of Wage-Rent-Profit Relations In A Two Sector Economy Using A Single Technique 6.7 172 Wage-Rent-Profit Relations In A Two Sector Economy With Two Different Techniques Available 6.8 172 Wage-Rent-Profit Relations In A Two Sector, Two Technique Economy, With Either Wages Or Rents Equal To Zero 6.9 176 Wage And Profit Relations In A Two Sector, Multi-Technique Economy 7.1 206 Income Expansion With A Lagged Multiplier 7.2 229 Income Expansion Path Of A Multiplier-Accelerator Model 8.1 242 Effect Of Rent Control On (i) Current, and (ii) Future Housing Supply, According To Neoclassical Price Theory B:Z: 249 Long Period Supply Curve 8.3 276 Relation Of Autonomous To Induced Investment In The Production Of Housing 8.4 320 Gentrification And The New Middle Class i x L i s t Of Tables  Number Page T i t l e I 174 D i r e c t i o n Of Switch In Technique As Q^l oo , For Various Levels Of r x 1 INTRODUCTION Theoretical urban analysis may be said to begin with Robert Park and the Chicago school of Human Ecology; and in that School's achievements and shortcomings may be discerned the origins of a l l subsequent developments in th i s f i e l d . Park attempted to show how an organic "superstructure", based on competition, and a c u l t u r a l "superstructure" were linked in a "symbiotic" ( i . e . , d i a l e c t i c a l ) fashion, via the concept of urbanism. It was the development of urbanism which was held to account for the development of modern c i t i e s ( c f . R.E. Park, 1932). There are clear p a r a l l e l s to be drawn here between Hegel's concept of the unfolding of world history, with urbanism playing the same role in the development of the c i t y as does the Idea in Hegel's v i s i o n of the history of world development. Consequently, while Park's work, and that of his colleagues and d i s c i p l e s (Burgess, Wirth, Zorbaugh etc.) provided many e x i s t e n t i a l l y authentic accounts of c i t y l i f e , they could not provide a sustainable theore t i c a l rationale for the phenomena they witnessed. The u n i f i e d approach of the Chicago School could not be maintained in subsequent years, with one t r a i n of thought leading off on an Arthurian quest for the d e f i n i t i o n of 2 urbanism, another replacing ecological theories of urbanism with economic ones and, f i n a l l y , yet another version seeking to uncover the moral order of the c i t y through a variety of exercises in sp a t i a l sociology. It is these l a s t two developments which are the p r i n c i p a l concern of this thesis, since i f the idea of Park as the Hegel of the c i t y i s accepted, urbanism as such i s c l e a r l y undefinable. Performing Marx's "double inversion" (Althusser, 1971) of Hegel's d i a l e c t i c of Park's d i a l e c t i c , we find not simply that the development of urbanism, but that the c i t y and therefore urbanism also, i s the manifestation of the economic organisation of those who created i t . In modern Western c i t i e s , the economic organisation of their inhabitants i s i n d u s t r i a l capitalism, and when we examine the economic (and sociological) contributions to urban analysis, we s h a l l do so only with such c i t i e s s p e c i f i c a l l y in mind. Since the demise of the Chicago School, economic and soc i o l o g i c a l approaches to urban analysis have pursued p a r a l l e l paths, i . e . , never once meeting. The economic theory employed by the descendants of the ecological school, neo-classical economics, stressed the sovereignty of the consumer and his a b i l i t y to make rational choices between housing consumption and the consumption of a l l other goods, given the constraint of the consumer's budget, as being responsible for the s p a t i a l pattern of the c i t y . The so c i o l o g i c a l approach, by contrast, stressed individual powerlessness in the face of the system. The individual had his or her choices made for them, either on the 3 basis of race or ethnicity via segregation, or on the basis of bureaucratic a l l o c a t i o n procedures, set either by mortgage finance i n s t i t u t i o n s , governmental housing agencies, or private landlords. However, just as "the system" allocates such power is not made at a l l clear. The two approaches to urban analysis, the economic and the s o c i o l o g i c a l , co-exist in a sit u a t i o n of "two solitudes", and the urban geographer, seeking to use both approaches in a synthesis of his or her own, would have to maintain an almost schizophrenic state of cognitive dissonance to do so. This thesis seeks to demonstrate that a way of synthesizing the economic and s o c i o l o g i c a l approaches to urban analysis can be found. It does so via an analysis of the various approaches employed in urban analysis. It begins with a thorough review and c r i t i q u e of neoclassical methods of analysis (chapters 1 to 4). This leads into a discussion of the various alternatives which have been proposed, notably the neo-Ricardiah and post-Keynesian approaches (chapters 5 to 8), and concludes (chapter 9) with a discussion of the implications for urban sociology of the theory developed in chapter 8. In th i s Introduction we b r i e f l y rehearse some of the p r i n c i p a l arguments contained in the thesis. The structure of chapters 1 to 8 broadly follows the course of the debate between the neo-classical school and i t s c r i t i c s . To begin with, the c r i t i c s of the neo-classical school were content merely with expressing doubts about the v a l i d i t y of the use of aggregate production functions to account for the 4 d i v i s i o n of the national income between wages, rents and p r o f i t s . Although t h i s debate was conducted over the issue of production, the neo-classical habit of treating consumers and producers in the same way means that the arguments conducted in the sphere of neo-classical production are capable of being translated into the sphere of neo-classical consumption. This i s demonstrated in the subchapter on u t i l i t y theory (chapter 1.1). In. the 1960's the debate moved onto an altogether higher planewith the publication of Sraffa's "Production of Commodities by Means of Commodities" (1960), Passinetti's revision of Kaldor's work on alternative theories of d i s t r i b u t i o n (Pasinetti, 1962; Kaldor, 1955), and the development of c a p i t a l vintage models (by, amongst others, Salter, 1962 and Jorgenson, 1966 - see Harcourt, 1972). Capital vintage models and their application to urban analysis are discussed in chapter 4. The c a p i t a l vintage model was the neo-classical response to the c r i t i c i s m s surrounding the use of the aggregate production function. These centred on the c a p i t a l concept used in t h i s model. Rather than using a single aggregate production function to analyse output and d i s t r i b u t i o n , the c a p i t a l vintage method aggregates time dated production functions. However, as the debates progressed, i t became clear that the problem lay not so much in the aggregation procedure used, but in the use of the production function i t s e l f . This was the conclusion of the Samuelson-Garegnani debate summarised in chapter 5.5. This debate was the culmination of the new avenues opened 5 up by Sraffa's work. Sraffa presented a model of an economy with heterogeneous c a p i t a l goods and a techripd-o^ i c a l l y determined surplus. In his model, a l l prices could be determined by reference to an invariant standard of value and no given technique of production was necessarily associated with any single value of the wage or p r o f i t rate. This was a direct challenge to neo-classical economics, which sprang from the i n a b i l i t y of c l a s s i c a l economics to measure changes in prices when methods of production change. In neo-classical economics, prices are set in exchange, not in production, d i s t r i b u t i o n of income i s determined by technology and the amount of the surplus produced is indeterminate. It is v i t a l for neo-classical economics therefore that any technique of production be associated with only one range of wages and p r o f i t s , otherwise technology alone cannot explain how a pa r t i c u l a r d i s t r i b u t i o n of income a r i s e s . Sraffa's model and those derived from i t have been dubbed neo-Ricardian, since they provided the solution to the problem which had prevented Ricardo from ever closing his system. Ricardo's work on the question of land rent, and the a f f i n i t i e s of his system to that of von Thunen, has led to an examination into the p o s s i b i l i t i e s of creating neo-Ricardian land use models (Scott, 1975,1976). The reasons why these models are l i k e l y to exhibit def icienc iei":ace examined in chapter 6. If technology did not determine d i s t r i b u t i o n of income, what did? Kaldor produced a model, revised by P a s i n e t t i , which 6 related d i s t r i b u t i o n of income and economic growth to investment behaviour, whether of c a p i t a l i s t s or of s o c i a l i s t planners. This model was developed out of Harrod's pioneering enquiries into the consequences of applying Keynes' ideas to the long period. In addition, they also argued that Keynesian theory ought more properly to be regarded as a dynamic extension of c l a s s i c a l theory rather than as a means of salvaging neo-classical theory from the morass in which the Great Depression had l e f t i t . Practitioners of Keynesian economics divide into post- and neo-Keynesians over their attitude to th i s issue. As the post-Keynesian theory became more f u l l y a r t i c u l a t e d during the 1970's, i t began to post a challenge to that hitherto unaffected part of neo-classical economics - general equilibrium theory. Pasinetti (1974, page 94) writes: In general equilibrium theory, traders with given stocks and tastes peform exchanges in markets such that every trader maximises his u t i l i t y . How traders come to be endowed with such stocks is not explained. At best general equilibrium (GE) theory can provide an analogy to an ephemeral period of economic a c t i v i t y . However, even i f the analogue comes out with the same results as that of the real s i t u a t i o n i t has been modelling, i t s t i l l does not explain how the r e a l world operates. The tragedy of the present world recession i s that i t has been created by people trying to f i t the world into the mould of a model such as t h i s . Chapter 8 discusses the adverse implications that post-Keynesian economics has for GE theory, p r i n c i p a l l y in terms of 7 i t s e f f e c t s on neo-classical price theory. However, GE theory i s unsuitable for the analysis of the space economy which characterises the real world on at least three counts. F i r s t , l o c a t i o n a l i n d i v i s i b i l i t i e s v i o l a t e the assumption of convex production and u t i l i t y functions. Figure 1.1 i l l u s t r a t e s t h i s point. This is a standard p a r t i a l equilibrium problem: the producer, or consumer, has a given budget constraint, B-B, and chooses that isoquant, or indifference curve, I-I, which i s just tangent to B-B. This point of tangency (Xj ,X2) gives the optimum combination and quantity of goods to be sold or bought. The problem GE theory sets i t s e l f i s to work out the optimum solution in n dimensions, i . e . , when there are n types of goods to be bought and sold. It i s clear that for t h i s procedure to work the iso-curve I-I must be convex downwards in order that there may be only one point of tangency. Should the iso-curve be non-convex, as i s the curve I'-I' , then corner solutions w i l l r e sult, with the optimium solutions being either ,0) or (0, . .Xj )• Clearly, in such a s i t u a t i o n , no equilibrium w i l l be either unique or stable. Consequently space i s always devoted in general equilibrium texts to showing that only convex iso-curves e x i s t . The proofs offered, however, tend to resemble St. Anselm's' Ontological Proof of the existence of God more than anything else. To conceive of perfection i s at the same time to demonstrate i t s r e a l i t y . General Equilibrium theory i s so perfect ("elegant") in i t s conception of the working of the I FIGURE 1.1: T r a d e - O f f P o s s i b i l i t i e Under Opposing Assumptions On I n d i v i s i b i l i t i e s 9 economy that i t necessarily en t a i l s the existence of convex iso-curves as guides to the conduct of a l l good, r a t i o n a l economic men. However, as Richardson (1977) and Artie and Varaiya (1975) argue, locational i n d i v i s i b i l i t i e s create inherent nonconvexities in both consumption and production. At any instant, a household, shop or unit of production can only be located at one place. It cannot trade off more of one location against less of another. The necessary existence of convex is o -curves i s l i k e that of most re l i g i o u s apparitions, which as Kregel (1980) puts it , are only seen i f believed. A second, related objection concerns the most fundamental of GE assumptions, namely knowledge of the existence of a l l markets. As Artie and Varaiya point out, i f "a r e t a i l e r is planning to set up shop in a c i t y " , By virtue of his locating at a certain point he automatically creates a new marketplace there, and excepting some t r i v i a l cases he w i l l not know the (equilibrium) price at his location of the commodities that he plans to trade. Hence his l o c a t i o n a l choice cannot be inferred from an Arrow-Debreu framework since i t cannot be a function of the equilibrium prices . Of course, after he has located the resulting prices could well be explained in an Arrow-Debreu framework, but his choice of location cannot. The question of the role of prices in conveying information to the economic decision-maker i s an important one, but before discussing i t , l e t us examine the way in which the c h a r a c t e r i s t i c s of a space economy vi o l a t e the assumptions of the GE approach. As we have said, one of the fundamental assumptions of GE methodology is the existence of a l l markets, 10 including markets for a l l future trades. Radner (1969) showed that i f any new markets are introduced at any l a t e r date, agents w i l l never have enough information to evaluate the f u l l set of alternative stategies that they might pursue. Agents need to ensure that the goods and services that they offer for sale may be sold at some positive p r i c e . In an economy with futures trading they need to be able to compute a l l possible outcomes of the pattern of sales and purchases for any set of prices which w i l l maximise their p r o f i t s given the price sets of a l l other goods and services. Furthermore the prices to be considered include not only a l l those possible now, but also a l l possible future p r i c e s . Even under the assumption that a l l future markets exist in the present the burden of information required by the GE approach i s severe. Once the p o s s i b i l i t y of spot markets coming into being in the future i s introduced, the burden of information is too severe, and the GE approach "breaks down completely" (Radner, 1969) in the face of i t . Agents w i l l never have s u f f i c i e n t information to evaluate the f u l l set of alternative stategies. A demand for l i q u i d i t y as a hedge against non-realisation of plans w i l l a r i s e . Furthermore, th i s demand for l i q u i d i t y w i l l i t s e l f be indeterminate. This finding i s of great significance for the advocates of Keynesian and post-Keynesian theory, which take as their s t a r t i n g point the c o n f l i c t between the need to commit resources to the future in order to produce goods, and the fear that the future w i l l not l i v e up to expectations; a c o n f l i c t whose state i s measured in 11 the degree of l i q u i d i t y preference at any time. Clearly, the fact of locati o n a l i n d i v i s i b i l i t i e s and the potential for new markets caused by the unique lo c a t i o n a l choices that they create is s u f f i c i e n t to create such a demand for l i q u i d i t y . However, there is also another reason why the assumption of the existence of future markets i s violated by the conditions of a space economy. This i s due to the fact that the space economy i s characterised by the existence of transport costs and economies of scale (Capozza and Van Order, 1977). The fact that a l l trades are accompanied by geographical transfers of goods and money means that there are transaction costs involved with every trade. Hahn (1973) confirms Radner's findings when transaction costs are introduced. Radner's a r t i c l e begs the question as to why new markets should be introduced i f agents are already equipped with a new set of stocks before trading begins. The answer is c l e a r l y that i f there are non-constant returns to scale in production, then any change in the l e v e l of productive a c t i v i t y would lead to a re-ordering of production (Kaldor, 1972) and consumption (Pa s i n e t t i , 1974) and a consequent appearance of new markets. However, production i s not considered within the framework of a GE model. It i s a model of a barter rather than of an i n d u s t r i a l economy. Stocks are given, not produced, and money i s not required to finance credit between the time of investment and the time of sale of the finished good, or as a hedge against f a i l u r e of expectations, brought on by changes in production, 1 2 rendering a technique obsolete. GE models describe in great d e t a i l the type of economies that Keynes set out to show in "The General Theory of Employment, Interest and Money" (1936) were irrelevant to the analysis of a modern c a p i t a l i s t economy. The questions posed by the existence of these objections to GE theory have ramifications that may not be immediately apparent. The question of the nonconvexities caused by locatio n a l i n d i v i s i b i l i t i e s means that the often expressed contention that economics can only deal with trade-offs, not with the establishment of p r i o r i t i e s , cannot be maintained. Locational i n d i v i s i b i l i t i e s mean that each location w i l l be associated with d i f f e r e n t sets of choices, ones which must be decided upon. The i n a b i l i t y of neo-classical economics to deal with t h i s problem are furtherdiscussed in chapter 1. Secondly, i f the composition of output and the pattern of demand are dependent on the scale of the economy, then c l e a r l y the decisions as to the l e v e l of gross investment to be carried out in any period w i l l be a major influence on the prices of commodities. GE theory assumes that a l l prices w i l l be established as the results of the interaction of supply and demand curves, and that these prices w i l l not only equate supply and demand in the present but also guide the a l l o c a t i o n of resources in the future. As the sections on housing supply and demand theory and the Keynesian c r i t i q u e of micro-economic price theory make clear, t h i s i s to place an impossibly heavy burden on the price system. 1 3 Furthermore, for the price system to operate in the manner assumed by general equilibrum theory, the supply and demand curves must be independent of each other. The conditions under which th i s i s possible are ones of perfect competition, with each firm so small in re l a t i o n to the t o t a l quantity of output that i t cannot affect the price, but with each firm capable of s e l l i n g an i n f i n i t e amount at that pr i c e . Not only are these conditions for firm structure and behaviour extremely unlikely even to be approached in r e a l i t y , but the conditions of the space economy, especially the existence of transport costs render a l l competition e s s e n t i a l l y imperfect (Chamberlin,1956). Further yet, i f demand i s affected by the scale of the economy, then conditions of supply c l e a r l y a f f e c t those of demand. Keynesian and post-Keynesian economics,by contrast, place the primary emphasis on the decision to invest. Beginning in this way, with production rather than exchange,they approach d i r e c t l y the conditions under which an i n d u s t r i a l economy operates, making i t immediately more relevant than trying to operate on an analogue of an i n d u s t r i a l economy based on production, by developing a complicated model of a barter economy, based on exchange. Keynesian economics began with the attack on the neo-c l a s s i c a l conception of the operation of the labour market as equating the supply and demand for labour through the intersection of the labour supply curve, measured in terms of amoney exchange rate, with the labour demand curve, measured in 1 4 terms of a technologically determined real wage rate. In fact, as Solow and S t i g l i t z (1968) pointed out, the wage rate clears the goods market, not the labour market. F u l l employment cannot be ensured by labour cutting i t s money wage demands as thi s w i l l only reduce demand for consumer goods and ultimately the demand for labour i t s e l f . F u l l employment can only be ensured by entrepreneurs' investment decisions. As Shackle (1972) says, to have used the methods of p a r t i a l equilibrium to analyse something so all-encompassing as the labour market was c l e a r l y to abuse the li m i t a t i o n s of this device. Restating the problem in general equilibrium terms does not solve the problem. As i s well known, Keynes knew l i t t l e of the works of Walras, casting his theory in terms of a c r i t i q u e of Say's Law, and believing that his theory would merely provide the guarantee of f u l l employment which would enable neo-classical analysis to f l o u r i s h again. Broadly speaking t h i s i s the position held by the neo-Keynesians, for example Samuelson and Pen. It took a long time for the r e l i s a t i o n to dawn that Walras' as well as Say's Law of Markets had been undermined. The General Theory, far from patching up neo-classical economics (despite what i t may have done for capitalism), in fact posed i t an unanswerable challenge. Walrasian economics l i e s at the basis of GE theory, which stands or f a l l s therefore with the v a l i d i t y of Walras' Law. Say's Law has i t that i f in equilibrium there are n commodity markets, and n-1 clear, then the nth w i l l clear also. Walras' Law i s that i f there are n+1 markets, of which the n+1th 1 5 is the market for money, then in equilibrium i f n clear, the n+1th, the market for investible funds, w i l l clear also. Both these Laws of Markets in fact " f a i l e d in the c r u c i a l test, on the labour market" (Kregel, 1973) during (and since) the Great Depression. As Kregel (1973) put i t , the price system is unable to co-ordinate decisions to invest at a le v e l which produced f u l l employment "for i t could neither provide the necessary information, nor produce effects cn expectations that might overcome th i s lack of information". Thus though the money market cleared as savings continued to equal investment, the direction of employment levels was down, not up. As the c a p i t a l vintage theorists recognised, one must be careful to make a d i s t i n c t i o n between ex ante and ex post re l a t i o n s in analysing economic systems. The ex ante estimates of supply and demand used before a firm makes a decision on whether to produce a commodity for sale are mental constructs only, examples of those "petty p o l i t e techniques, made for a well-panelled Board Room and a nicely regulated market... Liable to collapse (once) the practice of calmness and immobility of certainty and security suddenly breaks down" (Keynes, 1937). Ex post estimates of what the supply and demand schedules actually were w i l l confirm or question the accuracy of ex ante estimates and may be used in an evaluation of future strategy, but otherwise have no relevance in a planning context. It i s impossible to say whether a change in price r e f l e c t s a movement along, or a s h i f t in either supply or demand curve. To argue 16 with Muth and Rydell (1981) that the phenomenon of house sales increasing as prices r i s e represents a s h i f t outwards of the demand curve in response to expectations engendered by price signals i s no more than an ex post facto r a t i o n a l i s a t i o n of the causes. If sales also increase when the price f a l l s , then there i s no functional correspondence between expectations, price movements and volume of transactions. Although we might want to discuss the effects of lags in response to price movements, as we do in the section on multiplier-accelerator models (chapter 7.5), the fact remains that, as Pas i n e t t i (1974) says, "given any arb i t r a r y succession of points through time, one can also find an algebraic equation of s u f f i c i e n t l y high order that w i l l f i t the data p e r f e c t l y " . But achieving t h i s does not elucidate the operation of any general economic p r i n c i p l e s . The closeness of the f i t w i l l not be a r e f l e c t i o n of the economic theory underlying the equation used, but of the parameters of the curve i t is being f i t t e d to. Rather than supply and demand being equated through the operation of the price system, Keynesian and post-Keynesian theory regards t h i s as being accomplished through fluctuations in output. The role of prices is to act as an accounting mechanism, to ensure that the costs as established by the scale of the operation as given by the level of gross investment plus any given mark-up are met. As well as making out the general case for the post-Keynesian theory, chapters 7 and 8 are concerned with 1 7 establishing i t s a p p l i c a b i l i t y to urban analysis. However, Keynesian economics i s normally applied to issues of employment po l i c y , and i s commonly associated with p o l i c i e s of aggregate demand management via f i s c a l measures. How i s this relevant to an analysis of the influence of the economics of housing supply on location, and the parameters thus established, to the enquiries of urban sociologists? To begin with, there are a l o t of analogies to be made between the operation of the labour market and the housing market (see, for example, Capozza, 1980). Both deal with the r e l a t i o n between a nonproduced factor of production, labour in the one case, land in the other, with c a p i t a l , a produced means of production. More s p e c i f i c a l l y , employers supply vacancies to the job market, to be f i l l e d by workers; landlords and developers supply housing vacancies to the housing market, to be f i l l e d by households. In addition, no matter what one's views on the efficacy of the price system in general,clearly the housing market i s at least as encompassing in nature as the labour market. One would therefore be equally hard pressed to maintain the assumption of independence of the supply and demand schedules which would be necessary to allow the price s i g n a l l i n g mechanisms to operate unbiasedly. Secondly, i t is important to distinguish Keynes' theoreti c a l contribution from his analysis of the causes of the Depression and his prescriptions for a l l e v i a t i n g the unemployment of that era. As Keynes himself said, his theory 18 Does not offer a ready-made remedy as to how to avoid these fluctuations and to maintain output at a steady optimum l e v e l . . . Naturally I am interested not only in the diagnosis but the cure and many pages of my book are devoted to the l a t t e r . But I consider that my suggestions for a cure, which, avowedly, are not worked out completely, are on a d i f f e r e n t plane from the diagnosis. They are not meant to be d e f i n i t i v e ; they are subject to a l l sorts of special assumptions and are necessarily related to the p a r t i c u l a r conditions of the time. (Keynes, 1937) A l l that Keynes had to say, having i d e n t i f i e d the causes of unemployment as due to investment levels f a i l i n g to make up the gap between ef f e c t i v e demand and potential or f u l l employment demand, in his 1937 a r t i c l e was There is always a formula... Relating the output of consumption goods which i t pays to produce to the output of investment goods; and I have given attention to i t in my book under the name of the M u l t i p l i e r . The fact that an increase in consumption is apt in i t s e l f to stimulate t h i s further investment merely f o r i f i e s the argument. The unfortunate truth, however,is that there has been l i t t l e attempt to go beyond Keynes "avowedly" sketchy prescriptions for the a l l e v i a t i o n of depression-era unemployment. Yet as chapter 7.5 makes clear, the very fact of investment means that the parameters of the system a l t e r , and to understand the effects of investment, dynamic analysis is required. For Keynes himself the orthodoxy he was attacking was adequate in explaining the r e l a t i v e composition of goods which went to make a given l e v e l of output, but not in explaining what caused the output to be at that p a r t i c u l a r l e v e l . The issue of the composition of aggregate output however i s only non-19 problematical in a situation where an economy which was producing at f u l l capacity cuts back on output for whatever reason. The o v e r a l l l e v e l of u t i l i s a t i o n of plant would f a l l and the l e v e l of employment would f a l l with i t . Increase in demand for output through some f i s c a l incentive would then lead to increased employment by stimulating investment in wages alone. This i n c i d e n t a l l y is an important reason as to why marginal productivity does not regulate wages. The employer's wage b i l l consists not only of the cost of the labourer but the interest payable on the finance required to pay the labourer's wages between the time of commencement of employment and the r e a l i s a t i o n of the value of his or her output in sale (see Kregel, 1973). Another reason of course is that the marginal productivity of labour is measurable only i f c a p i t a l i s homogeneous both with i t s own means of production and with the wage good (see chapter 1 below). If , however, investment is in wages only, then the question of the composition of output never a r i s e s . As no investment i s being made in plant, the composition of output i s given. The Depression originated in a f a i l u r e of f i n a n c i a l confidence and occurred very suddenly. It was not a result of a long period of f a l l i n g i n d u s t r i a l productivity, such as has plagued the UK economy in the post-war years. Given the problems posed by the Depression, aggregate demand management through f i s c a l incentives was the appropriate response for that time. Domar was the f i r s t to point out that net p o s i t i v e investment i s as l i k e l y 20 to change the pattern of output as to maintain i t . As soon as we allow investment to change the pattern of output, then gaps between e f f e c t i v e and potential demand can arise through investment being of the wrong type, and not simply the wrong amount (for the maintenance of f u l l employment, see Shackle, 1972). The t r a d i t i o n a l "Keynesian" remedies for unemployment are not so much wrong as anachronistic. This is because no-one has gone very far beyond Keynes or Domar in investigating the issue of the composition of investment (though see Shackle,1972 and Robinson,1971). The implication of investment being of the wrong type i s that the pattern of employment that i t generates w i l l be of the wrong, type also. Herein l i e s the clue as to how the concept of aggregate demand, heretofore regarded as wholly macro-economic in character/may be disaggregated right down to the personal l e v e l . Increases in the correct type of investment not only raise the l e v e l of e f f e c t i v e demand, they raise labour productivity also. Generally speaking, the history of labour productivity over the last century has been one of increasing output and wages per head, despite decreasing hours of work. That people can produce more and receive higher wages while working less hours i s because of increases in the amount of c a p i t a l invested per worker. This applies to doctors, lawyers and plumbers as much as to, say, auto workers or a g r i c u l t u r a l labourers. The training of doctors, lawyers and plumbers are investments in time.and money, reflected in the r e l a t i v e l y highwages they 21 receive. The higher wages received by auto workers as compared to a g r i c u l t u r a l labourers is a r e f l e c t i o n of the amount of c a p i t a l invested in providing that worker with a job, whether that investment i s d i r e c t , that i s , in training, , or i n d i r e c t l y as well, v i a plant and machinery. However, .effective demand per household is not simply a r e f l e c t i o n of any job related investment, but may include any other investment made in that household for any s p e c i f i c purpose. Thus the amount a mortgage finance i n s t i t u t i o n i s w i l l i n g to invest in a household w i l l largely determine their e f f e c t i v e demand for owner occupied housing. The explanation of e f f e c t i v e demand in terms of personal level s of investment links the behaviour of mortgage finance i n s t i t u t i o n s to individual demand for housing. The concluding sectons of chapter 6 are devoted to an elaboration of t h i s idea. They show how the Keynesian analysis of the relations between l i q u i d i t y preference, the interest rate, the volume of investment and output are applicable d i r e c t l y to the case of housing supply. Not only that, but they also show how the conditions which prev a i l in the housing market are no di f f e r e n t in kind from those which p r e v a i l in almost any other non-a g r i c u l t u r a l market, namely that the product i s heterogeneous and i t s market i s dominated by exis t i n g stocks, with one major exception. The conditions of financing housing are unfavourable compared to other f i n a n c i a l ventures. Mortgage i n s t i t u t i o n s borrow "short", r a i s i n g finance p r i n c i p a l l y in the form of 22 demand deposits. They of necessity lend "long", using the house as security. Their asset / l i a b i l i t y structures are therefore rather exposed, with their assets t i e d up in i l l i q u i d properties, and their l i a b i l i t i e s being in the form of short term loans (Charles, 1977). To compensate for t h i s , the rate of interest on mortgage finance must be a few points higher than the rate of interest on other investments. This, i t is argued, has the eff e c t of lowering e f f e c t i v e demand in thi s sector. Coupled with an analysis of how the le v e l of e f f e c t i v e demand i s l i a b l e to vary d i r e c t l y but nonlinearly with income (cf. Kregel, 1981), an explanation can be offered as to why housing shortages tend to be persistent and to affe c t those on low incomes more severely than other groups. As low incomes are associated with low level s of e f f e c t i v e demand, thi s implies that those with low incomes are also employed, i f at a l l , in sectors with low levels of investment per worker. In general, firms in such sectors have f a i r l y high geographical mobility associated with ease of entry into and exit from their sectors. Low income, however, tends to act as a barrier to geographical mobility of labour. Low income workers therefore tend to get l e f t behind when the pattern of employment changes. They can become "trapped" by the housing system as a res u l t . Urban growth i s nonetheless a result of investment in the building stock, however th i s may affe c t the labour force. Chapter 8 looks at how the built-up area at any time offers a 23 pattern of opportunities and constraints, which w i l l at one time lead investors to concentrate on "greenfield" development on the outskirts of the c i t y , and at other times to go for redevelopment or renovation of inner c i t y stock. As well as d i f f e r e n t locations, the financing of d i f f e r e n t types of tenure are discussd. Examples are given of instances where the one type of development w i l l occur in preference to either of the others. This interaction between the investment opportunities as perceived by potential investors and the constraints posed by the b u i l t form of the c i t y (and by those with control over the disposal of the properties that comprise-this b u i l t form, which includes municipal planning departments as well as property owners) is seen as being a d i a l e c t i c a l one. For this reason, and because the Keynesian d e f i n i t i o n of the long period i s no more than an aggregation of short periods (and thus contains no c h a r a c t e r i s t i c s not present in the short period), patterns of growth are not seen as being amenable to modelling by mathematical methods. Chapter 9 attempts to draw out the implications of the forgoing discussions for urban s o c i a l theory. It re-iterates the contradictions between conventional urban economic and s o c i a l theory, and then shows how the p r i n c i p l e s outlined in chapter 8 can provide an economic context for urban s o c i a l theory. The power of urban managers and gate-keepers and the a b i l i t y of various s o c i a l groups to practise segregation, for example, are functions of market conditions. The p r i n c i p l e s outlined in 24 chapter 8 may also be used in a c r i t i q u e of urban s o c i a l theory. The two p r i n c i p a l theories of class structure used by urban s o c i a l theorists are based on Marx and Weber respectively. Widely regarded as being a n t i t h e t i c a l in nature, chapter 9 shows how they can be reconciled, and also how the theory of personal e f f e c t i v e demand as outlined in chapter 8 can account for the various types of Weberian cl a s s . Also, the debate in Marxist c i r c l e s as to which c h a r a c t e r i s t i c s of housing (as, for example, either necessity or investment good) ought to be taken into account for the purpose of analysis, and whose judgement as to the importance of those c h a r a c t e r i s t i c s i s the relevant one (that i s , the analyst's or the inhabitant's) i s c r i t i c i s e d . Such debates treat housing on a subjective basis, as an object to be invested with one value or another by the observer or actor in the housing scene. A treatment of housing consistent with an objective standard of value should begin by regarding i t as a commodity, to be produced, l i k e any other commodity. Only after having examined the conditions of production, and the d i s t r i b u t i o n these conditions are l i k e l y to create should any special properties that housing might have be investigated. Assigning properties to housing on an ad hoc basis according to the preferences of the analyst is a procedure in which the assulptions merely e n t a i l the conclusions. Analyses which start by regarding housing as some sort of peculiar commodity inevitably w i l l f a i l to understand i t s li n k s to the wider economic and s o c i a l context. 25 Comparing housing to other "necessities" or investment goods reveals only one c h a r a c t e r i s t i c peculiar to i t in any s i g n i f i c a n t degree, and that i s location. Locational i n d i v i s i b i l i t i e s , generally inimical to microeconomic analysis as they are, have a special significance in the case of housing. They form an element of class structuration as well as magnifying "existing differences in income and l i f e chances" ( B e l l , 1977). Keynesian economics has suffered in the past from i t s lack of any "micro" content. However, i t s assumptions are not undermined when the c h a r a c t e r i s t i c s of a space economy are considered. The p r i n c i p l e s outlined in chapter 8 indicate how Keynesian theory may be applied on a micro basis. Keynesian and post-Keynesian economics may then be used to provide an account of how a housing shortage may develop and p e r s i s t . This i s a necessary condition for power to be exercised by urban gate-keepers and the l i k e . Chapter 9 concludes by arguing that the use of post-Keynesian theory is consistent with the requirements of a class-based urban sociology. It thus.provides a framework in which urban growth, location and a l l o c a t i o n may be discussed, and, hopefully, a means by which the synthesis which eluded Park and his colleagues might be achieved. 26 1 NEO-CLASSICAL APPROACHES TO URBAN PROCESSES: REVIEW OF METHODOLOGY Neo-classical approaches to urban modelling have by and large followed the production function methodology. Unlike the general equilibrium approach, which emphasises the role of supply and demand in determining prices, the production function application determines prices from the technical conditions of production, as expressed in the production function i t s e l f . The production function methodology is symmetrical with the analysis of consumer choice. Producers maximise output by the optimum purchase of inputs: consumers maximise u t i l i t y by the optimum purchase of goods; in both cases, the optimisation being with respect to some budget constraint. Both u t i l i t y and output have similar metaphysical q u a l i t i e s , as w i l l be shown. Muth (1969) pioneered the production approach, and Solow (1973) the u t i l i t y approach (though see also Alonso, 1964) in urban economics. The major technical problem in neo-classical economics i s aggregation and how t h i s i s to be achieved without prohibiting the use of d i f f e r e n t i a l equations to describe marginal rel a t i o n s . This w i l l be discussed with respect to u t i l i t y theory and to production theory. 27 1.1 U t i l i t y Theory The theory of u t i l i t y maximisation holds that consumers try to maximise some psychological quality c a l l e d u t i l i t y by d i s t r i b u t i n g their income in such a way that the r e l a t i v e u t i l i t i e s derived from the goods purchased i s in the same ra t i o as the r e l a t i v e price levels of these goods. As an example of a constrained maximisation problem, i t uses the Lagrangian method of solution: MaxU = tKXj ... X j Subject to lp. X. = rl Where U i s u t i l i t y , X t i s any good, Pt i s the price /unit of that good and M is the individual's t o t a l budget. For s i m p l i c i t y assume n=2, then the problem becomes MaxU = U ( X i f X 2 ) Subject to p1X1-+p2Xz - M Form a Lagrangian where^jl(mu) i s the Lagrangian m u l t i p l i e r . Thus max/ = maxU as yU(h " P ^ - J ^ x J = 0 A condition of maximisation i s that the f i r s t order p a r t i a l derivatives of (^ X/^ X^  = X^) equal zero:; = U - ;Up = 0 4 = U-yUp = 0 28 However these conditions are s a t i s f i e d whenever the function has a maximum, a minimum, or a saddle—point (a maximum in some directions, and a minimum in others). The second order conditions are that < Q for a l l variables ( t J - X ^ X ^ j l O . This i s s a t i s f i e d by having the determinant ( Q ) of the matrix of second order p a r t i a l s greater than zero ( Sil'berberg, 1978 ) D = t l -L +-11 f^Xfi. i l l ^11 r^jx t^i JJ. "^-ju ^-p- ^p-ji- - R - R 0 > 0 The second order conditions, that J) 4 ® ( i f D < 0 ' t h e solution i s a minimum ), are also the guarantee that the system of simultaneous equations represented by the f i r s t p a r t i a l s o f ^ have a d e f i n i t e solution. The system of f i r s t p a r t i a l s of j£ can be represented in a matrix, known as a Jacobian; for t h i s system to be determinate, the determinant of the Jacobian, J, must not be equal to zero. A non—rigourous proof of this i s that in general the solution of a set of simultaneous equations i s achieved through the application of Cramer's rule: for any system of simultaneous equations 1 % « * \ /*A / M Cramer's rule states that for any variable, 29 where f\ is the determinant of the matrix of a .'S• Obviously, i f t h i s i s zero, X|_ w i l l be undefined. The geometric interpretation of a zero determinant i s that the system of equations i t describes are a l l p a r a l l e l , hence no point exists at which a l l these equations reach a mutual maximum. The guarantee that J i s not in fact equal to zero i s provided by the i m p l i c i t function theorem (Silberberg 1978, p. 186): "the i m p l i c i t function theorem... says that i f the determinant of the f i r s t p a r t i a l s of a system of equations i s non—zero, those equations can be solved l o c a l l y ( in p r i n c i p a l — not perhaps easily) for those variables being d i f f e r e n t i a t e d as e x p l i c i t functions of the remaining variables (here the parameters )of the system. The determinant (Q) i s such a determinant and i s non—zero.... by the s u f f i c i e n t second—order conditions." The determinant [] i s the determinant of the second p a r t i a l s of^_, but also the determinant of the f i r s t order p a r t i a l s of J. Therefore the s u f f i c i e n t conditions for the maximisation of that J ) be non—zero, also guarantee the existence of a determinate solution for J, i . e . , that J also i s non—zero. It can be assumed therefore that a solution to the system of f i r s t order p a r t i a l s exists, and can be written as The parameters are the prices and money income. The marginal relations from which these e x p l i c i t solutions are 30 derived are not observable, but the demand relations ( X^ - ) relate to observable variables ( [D^  p>, ) and are therefore p o t e n t i a l l y useful. If they are substituted into (JCX^X^K an indi r e c t objective u t i l i t y function i s created U'lft, fc, M) = UlxTlp,, R , M) x f l f i , R, M)) a function only of the parameters, prices and money income. The function (J gives the maximum value of u t i l i t y for any given prices and money income since the values of X ^ X j which maximise thi s subject to the budget constraint are the ones which have been substituted into I J O ^ X^) . S i m i l a r i l y , an indirect objective production function i s a function only of factor prices and revenue, and maximises output for any given set of prices and revenues. From the f i r s t order conditions thus r^r •= — , as suggested. The r a t i o of the marginal u t i l i t i e s equals the r a t i o of the prices in respect of X x and X^• However, the most important aspect of t h i s analysis for the present discussion i s that the demand curves X* are homogeneous degree zero in prices and money income, that is X i ^ P j , tp f M) - X * (£ , J> ,M),since i t is r e l a t i v e prices only that are important in deciding consumer purchases. The problem MaxU =. U ( X , , Xz) Subject to fcPjX;+ tp x X 2 =:tM 31 is i d e n t i c a l to Max U - (jUi x j Subject to p ± X d + P a X 2 = M The demand curves are the e x p l i c i t solutions of the f i r s t order conditions for the maximisation of j[_ ( and ( J a l s o ) . If they are homogeneous of degree zero, then the u t i l i t y function from which they derive must be homogeneous of degree one, or is homothetic and can be transformed into such a function without loss of generality. A homothetic function, H, is not homogeneous i . e . , H ( t X 1 , t X z ) ^ t f(H) w ^ e r e ^ * s a n ¥ degree of homogeneity, but i t does share the property with homogeneous functions that the ra t i o of i t s f i r s t p a r t i a l s = a constant, the same h constant, in fact, as i s equalled by the f i r s t p a r t i a l s , *•], , "9. of the homogeneous function from which i t is derived. Homothetic functions must be monotonic transformations of the o r i g i n a l function, which can always be treated as i f i t were homogeneous degree one. If h - t\(y)' homogeneous degree r , then [h (y)] ^  i - s homogeneous degree one. Therefore i f H(y) = F[h(y)] ' t n e n F c a n be regarded as a composite function, the f i r s t performing the operation \f , and the second whatever transformation i s required to produce H. For u t i l i t y theory, the property that the r a t i o of f i r s t p a r t i a l s remains unchanged, means that the shape of the (isoquants) indifference curves do not a l t e r as income levels get higher (as the indifference curves s h i f t 'North—East' 32 on the graph). A l l the indifference curves are r a d i a l expansions or reductions of each other, and the locus of a l l tangencies of the indifference curves to the budget constraints (always p a r a l l e l ) i s a straight l i n e (Fig.1.1). In u t i l i t y theory such a locus is termed an income consumption path, or Engel curve. The fact that under th i s approach, a l l Engel curves are straight l i n e s is fortunate since Brown and Deaton (1968) report that straight l i n e Engel curves for a l l consumers i s a necessary and s u f f i c i e n t condition to permit consistent aggregation of individual consumer demand. The implications of a straight l i n e Engel curve are f i r s t that the pattern of consumer expenditure does not vary with increasing income. They buy more of everything, but nothing new. A consequence i s therefore that consumer goods are assumed to be i n f i n i t e l y d i v i s i b l e . This a problem which occurs also in r e l a t i o n to production aggregates. One other c h a r a c t e r i s t i c of the u t i l i t y maximisation method described here i s the interpretation of the Lagrangian m u l t i p l i e r . This can be ascertained from a consideration of the Lagrangian i t s e l f . If (J i s the indirect u t i l i t y function, then i t s choice variables are the parameters (p^f^M). Its rate of change with respect to a parameter is independent of any effect that parameter might have on the e x p l i c i t demand functions X * . This i s because the re l a t i o n between X A , X 2 a r e fixed at X * ( p i , |>, M)and xt(pi-> M) and these are homogeneous degree zero (Silberberg, 1978 p. 168). The rate of change of j£_ with respect to a parameter change is quite straightforward FIGURE.1.1: Income Consumption P a t h s Or E n g e l C u r v e s 34 s i n c e by d e f i n i t i o n X 1 5 X 2 must be f i x e d t o p e r m i t an e x a c t l y c o r r e s p o n d i n g .increase i n U • U t i l i t y , t h e o r y i s a c o n t r o v e r s i a l s u b j e c t , m a i n l y because of the p l a c e i t h o l d s a t the c e n t r e of n e o - c l a s s i c a l t h o u g h t . The u n d e r l y i n g p r i n c i p l e o f , f o r example, g e n e r a l e q u i l i b r i u m t h e o r y i s t h a t of u t i l i t y m a x i m i s a t i o n s u b j e c t not t o a money c o n s t r a i n t , but t o the t o t a l s c h e d u l e of goods and s e r v i c e s t h a t the agent has a v a i l a b l e f o r o t h e r goods and s e r v i c e s . I t i s here t h a t n e o - c l a s s i c a l economics' i d e o l o g i c a l b i a s e s a r e most c l e a r l y r e v e a l e d . For example, S i l b e r b e r g d i s c u s s e s the s u b s t i t u t i o n p o s t u l a t e , t h a t goods w i l l be t r a d e d — o f f f o r each o t h e r i n t h e s e terms: The s u b s t i t u t i o n p o s t u l a t e i s an e x p l i c i t d e n i a l of the ' p r i o r i t y of needs' f a l l a c y . P o l i t i c i a n s and p r e s s u r e groups are f o r e v e r u r g i n g t h a t we r e — o r d e r our p r i o r i t i e s i . e . , devote more r e s o u r c e s t o t h e goods they v a l u e more h i g h l y than o t h e r s . W h i l e i t i s u s e f u l f o r such groups t o t a l k of needs and p r i o r i t i e s , i t i s f a l l a c i o u s f o r economists t o do so. The n o t i o n of a ' t r a d e — o f f i s i n c o n s i s t e n t w i t h one good b e i n g ' p r i o r ' t o a n o t h e r i n consumption" ( S i l b e r b e r g , 1978, p. 220) d i f f e r e n t i a t i o n w i t h r e s p e c t t o some v a r i a b l e , here . Thus so t h a t i s the m a r g i n a l u t i l i t y of money. A few pages l a t e r he w r i t e s ( op. c i t . p. 228, ): 35 "the demand c u r v e s a r e independent of any monotonic t r a n s f o r m a t i o n of the u t i l i t y f u n c t i o n : i . e . , they a r e independent of any r e l a b e l l i n g of the i n d i f f e r e n c e map. T h i s p o s i t i o n s i m p l y r e i n f o r c e s the n o t i o n t h a t o n l y exchange v a l u e s m a t t e r . A l o n g any i n d i f f e r e n c e c u r v e , the s l o p e measures the t r a d e - o f f s t h a t a consumer i s w i l l i n g t o make w i t h r e g a r d t o g i v i n g up one commodity t o get more of a n o t h e r . These m a r g i n a l e v a l u a t i o n s of goods a r e the o n l y o p e r a t i o n a l measures of v a l u e ; i t m a t t e r s not one w h i t whether t h a t i n d i f f e r e n c e c u r v e i s l a b e l l e d as 10 u t i l e s , or 1 0 , 0 0 0 , or 1 0 1 ° u t i l e s . I t i s the s l o p e , and o n l y the s l o p e , of the l e v e l c u r v e t h a t m a t t e r s f o r v a l u e and exchange, not some index of s a t i s f a c t i o n a s s o c i a t e d w i t h any g i v e n consumption bundle. In f a c t , i t i s i m p o s s i b l e t o t e l l whether a consumer i s p l e a s e d or d i s p l e a s e d t o consume a g i v e n commodity bun d l e . I f t h e s e a r e the o n l y goods over which he or she has t o make d e c i s i o n s , the exchange v a l u e s do not i n any way r e f l e c t whether the consumer i s e c s t a t i c or m i s e r a b l e w i t h h i s or her l o t . " I t i s no wonder t h a t i t i s i m p o s s i b l e t o t e l l whether a consumer i s p l e a s e d or d i s p l e a s e d t o consume a g i v e n commodity bu n d l e , s i n c e the q u e s t i o n of p r i o r i t y of needs i s e x c l u d e d from c o n s i d e r a t i o n . I f such p r i o r i t i e s were known then the a n a l y s i s of the consumer's c h o i c e would have some meaning s i n c e then the c o n t e x t of the c h o i c e would be known. The n e o - c l a s s i c a l paradigm i s t h a t of economics as an a n a l y s i s of c h o i c e , y e t the most im p o r t a n t a s p e c t of c h o i c e , t h a t of the c h o i c e of one o b j e c t i v e se t of c i r c u m s t a n c e s over a n o t h e r , i s d e n i e d c o n s i d e r a t i o n , w i t h the r e s u l t t h a t t h e o n l y a s p e c t of c h o i c e which i s c o n s i d e r e d i s i n h e r e n t l y n o n — i n t e r e s t i n g . To summarise S i l b e r b e r g , the making of c h o i c e i s u n s c i e n t i f i c ; o n l y the d i s p a s s i o n a t e a n a l y s i s of c h o i c e i s u n s u l l i e d by p a r t i s a n i n t e r e s t . However, s i n c e the c o n t e x t i n which c h o i c e i s made i s beyond the purview of the economist, the i n t e n t of the a n a l y s i s i s u n c l e a r , because i n a 36 given s i t u a t i o n , the choices any rational individual w i l l make are s i m i l a r i l y given. If i t cannot be stated whether a consumer is s a t i s f i e d or d i s s a t i s f i e d with a given commodity bundle, there is l i t t l e point in finding out in what proportions the commodities are consumed. Since neo-classical economics abandoned the notion of absolute u t i l i t y , so that the choice between alternative situations could be investigated, i t has become a t h i n l y disguised j u s t i f i c a t i o n of the status quo. It asserts i m p l i c i t l y that because choice is being studied, choice in the economic system therefore exists to be studied. The response of neo-classical economics to such c r i t i c i s m i s to assume that a l l consumers have perfect information, so that there are no alternative consumption bundles, so every substitution p o s s i b i l i t y i s known. However i f the same bundle of goods and services i s under consideration by a l l consumers, then should income d i s t r i b u t i o n be unequal, only the richest individual or group of individuals would be able to achieve an optimum position (that being the maximum u t i l i t y which can be derived from that bundle of goods and services) and everyone else would be in a suboptimal position by comparison. To a l t e r that s i t u a t i o n , however, would c o n f l i c t with requirements for the overall,,Pareto, optimality, that no—one should be able to make themselves better off without making some—one else worse of f . To base a theory of consumer choice on an analysis of the behaviour of perfectly informed individuals of equal income with 37 s t r a i g h t - l i n e income consumption paths, so that a l l individuals' patterns of purchases are i d e n t i c a l and unaffected by income, suggests that neo-classical economics considers consumption in the aggregate form to be treatable as though the aggregate consumption i s performed by an aggregate consumer, who consumes one a l l purpose commodity, or commodity bundle, which never a l t e r s regardless of how much or how l i t t l e i s consumed, i . e . , there are no substitution e f f e c t s due to change in income. It is d i f f i c u l t to see how this i s relevant to an analysis of any actual economy, but i t s use does imply that what differences in income there may be, they are not that important. Like the problem of choice, t h i s unanalysed assertion about the p r a c t i c a l importance of differences in incomes does betray a consistent ideological bias in favour of the system that claims to provide of freedom of choice, namely free enterprise capitalism. 38 1.2 Production Theory Three production functions have been reported in the urban economics l i t e r a t u r e , Cobb—Douglas, constant e l a s t i c i t y of substitution, and variable e l a s t i c i t y of substitution. The constant e l a s t i c i t y of substitution and variable e l a s t i c i t y of substitution both reduce to the Cobb—Douglas function under certain conditions, but they are not compatible with each other (Revankar, 1971). We sha l l review these functions in turn. The Cobb-Douglas function has a beguiling s i m p l i c i t y about i t , which has made i t the most popular of a l l production functions. It can be written in the form where Q = A K L!" tfhere Q= output, f( = c a p i t a l , {_ = labour, L\ , c* are parameters Or in the form a - A k* This intensive, log-linear form i s useful in econometric exercises. In both cases gives the share of c a p i t a l in value of output. This is because c a p i t a l ' s share i s given by fK\/Q» where T is the p r o f i t rate. Since, under the neo-classical assumptions, the p r o f i t rate equals the marginal productivity of c a p i t a l , either by accident or design (whether or the marginal productivity of c a p i t a l i s the independent variable i s a matter of debate among neo-classical economists, as i s the rel a t i o n between W the wage rate, and the marginal productivity of 39 labour, (Harcourt, 1972), l.e., r ~-6(Xjr) K' a n <^ c a p i t a l ' s share is given by the e l a s t i c i t y of output with respect to c a p i t a l , . Capital's contribution to output is ref l e c t e d in i t s share of the pie. This share i s measured by oc, known therefore as the d i s t r i b u t i o n parameter. The derivation for both the standard and intensive versions and the rel a t i o n between the two is given below: oc , |-ex Q = AK L. q = \ = A - K " t = A ( \ ) % A k * bK Inq = In A <• «Alnk = o«A(KU*" dlnq dink dink =-AK L' ding k dq Q.E.D. "dink " qdk r * Q.E.D The Cobb—Douglas function i s homogeneous of degree one, i . e . , xQ = AlxKHxl)1-* Its marginal products are therefore homogeneous of degree zero i . e . , XK \ _ 40 Because of t h i s , the Cobb-Douglas functional form i s important also in u t i l i t y theory. It i s used by Solow (1972) in the following form: Max (J(c,S) = '°q S + CI-K) loo C where S - residential space 0 0 C - consumption of C.CI other qoods pr=: renf atcUstomce r subject to y n c+p r S r + t r t r - transport costs at r (Were ' fc s distribution paramete rj The interpretation of K i s that i t i s the income e l a s t i c i t y of housing demand. One c h a r a c t e r i s t i c of the Cobb-Douglas form i s that i t i s m u l t i p l i c a t i v e , meaning that production cannot be carri e d out by land alone, that i s , without the aid of c a p i t a l . S i m i l a r i l y , both housing (S), and the composite good (c) must be consumed to achieve any u t i l i t y . This property is not shared by either the constant e l a s t i c i t y of substitution or variable e l a s t i c i t y of substitution function. These functions can best be introduced by a discussion of the e l a s t i c i t y of substitution (Silberberg, op. c i t , p. 313) Consider a Cobb—Douglas production function r» L- . A cost minimising firm s a t i s f i e s the f i r s t order conditions of the Lagrangian ^ ~ r K + W L+ A C y - § ( r \ > L ) ) where /\ i s the lagrangian m u l t i p l i e r . In this case, the f i r s t order conditions are 41 Upon d i v i s i o n equations 1.2.1 and 1.2.2 y i e l d K_ = _ o c _ w. U± 1.1.5 L i-<* r The capital—labour r a t i o can be therefore be considered as a simple function of the wage rate. For mathematical convenience, the left—hand side of equation 1.2.3 i s re—arranged to give: \ •. • , r where ' k = % » * = 1 P = r/* " The rate of change of with respect to to change in w / r i s given by Although th i s i s a measure of s u b s t i t u t a b i l i t y , the more frequent measure i s the dimensionless e l a s t i c i t y analogue tf _ _ dJi/Ji - dk_ dP/P k ' dp The minus sign i s added to make the measure p o s i t i v e . Applying th i s to the Cobb-Douglas case gives k p1- k " since 0.= kp from 1.2.4. Another way of defining 6 in the Cobb-Douglas case i s 6 ZL — . i l k - _oc_ w L / g I _ oc 6 L/<S0. remembering the discussion of ot as the d i s t r i b u t i o n parameter. Another way of defining o" i s to measure how fast the r a t i o of 42 the two inputs changes when the marginal rate of substitution changes. This is given by: L/K d( j L/f K) (where i s the p a r t i a l derivative of f(K,C) with respect to U K ) The solution of which for linear homogeneous functions i s ' - i t ->^ can also be related to the c r o s s — e l a s t i c i t y of factor demand: €w.= 3w-K- ' a n d 6 = f j ' f ' d t ' (Silberberg, 1978, p. 316) as - r — = ? = -—- i n the Cobb—Douglas case, t h i s gives ~ 1 c — 1 c ~ l-<x fcKL _ <x LK However, since 6= 1 in the Cobb—Douglas case, t h i s implies that € = 1 - o t > £ • = <* and therefore This indicates why the Cobb—Douglas function has been so popular in empirical work; a l l the technological c h a r a c t e r i s t i c s of interest can be expressed in terms of very e a s i l y computable parameters, since lnq^=lnA+<xlnk i s very easy to formulate as a regression equation. Also, in u t i l i t y theory, oV gives the price e l a s t i c i t y of demand. 43 Where the production function i s not Cobb—Douglas but If production cannot necessarily be characterised by a Cobb—Douglas function, but the ra t i o of factor inputs continues to be a function of the ra t i o of their wage rates, then 6 w i l l be a constant and characterised by Thus, a l l homothetic production functions have "the property that the capital—labour r a t i o i s proportional to the wage rate raised to some power, that power being the negative of the e l a s t i c i t y of substitution" ( Silberberg, p. 319). Rewriting this r e l a t i o n in terms of the o r i g i n a l variables yields taking roots OL = C P > 0 For convenience write OL = °V(i_cx.) ; as varies between 0 and 1-, CL varies between 0 and oo , thus no generality i s l o s t . Separating variables gives homogeneous degree one, theorem. There are two solutions to th i s s i t u a t i o n , depending on 44 whether 6 i s equal to one or not. Case 1, 6 =1, integrating... 1.2.5 ot f i e . = - o - < > o ( ¥ = + *>gW Where ' fc^e constant of integration, can be any function of J, since y was held constant in determining the slope of 6K /dL . Thus Or < J ( L J ) If Oj(y^ is monotonic, then there must exist a homothetic function y =. Where F i s the inverse function of (J , and a function of a function, homogeneous degree one, so that a function also exists \j = AK L Homogeneous of degree 5 , r e c a l l i n g the discussion of the re l a t i o n between homotheticity and homogeneity. Where 5=1, the Cobb—Douglas function r e s u l t s . Case 2, 6 t 1, we begin again by integrating... -.1.2.5 Giving Incorporating the factor % +1 into CJ , the constant of integration, and writing p.- , and re—arranging gives Again from the discussion on homogeneity and homotheticity, i f Cj(vj) i s monotonic, there must exist a homothetic function 45 VJ = r^K' p T( i-oc)L' p ) And, derived from t h i s , a function homogeneous degree one, y = fl(c*K'P + ( i - < * ) L " " p ) ^ 1.1.6. Since i f tj - , £ = f\ i , and can be written in the form F = Ax , F= i s homogeneous degree one. Equation 1.2.6 is the constant e l a s t i c i t y of substitution (CES) production function where 6 f- 1 , but can vary between 0 and i n f i n i t y as varies between -1<p<+°0. The variable e l a s t i c i t y of substitution function to be described below was introduced by Revankar(1971), and i s used by Kau and Lee(l976) in an urban application. After reviewing attempts to generalise the constant e l a s t i c i t y of substitution function, which founder on the complicated formulae for 6 that they require, which present d i f f i c u l t i e s of estimation, he comments that in the meantime a function of 6 might be found that permits a Cobb-Douglas generalisation. There have been a number of attempts to do this but they produce complicated functional forms for 6 . He considers a generalisation corresponding functional form of 6 , namely: <5 = 1 •+ (3 — p = a parameter ' Thus deriving his production function from a choice of 6 rather than thinking up a production function form and deriving a formula for 6 from i t . This p a r t i c u l a r choice of a form of is a special case of the general form 46 error term And was chosen " i n o r d e r t o impart economic meaning t o the parameter" (Revankar, 1971). However, where ^ 1 , a c o n v e n i e n t f u n c t i o n a l form f o r a v a r i a b l e e l a s t i c i t y of s u b s t i t u t i o n f u n c t i o n has not y e t been found. I t can be shown t h a t where jt 1 , t h i s g e n e r a l form g e n e r a l i s e s t o a c o n s t a n t e l a s t i c i t y of s u b s t i t u t i o n p r o d u c t i o n f u n c t i o n . By c h o o s i n g (3A s i , t h e r e f o r e , the p o s s i b i l i t y of p r o d u c i n g a v a r i a b l e e l a s t i c i t y of s u b s t i t u t i o n f u n c t i o n which i n c l u d e s the c o n s t a n t e l a s t i c i t y of s u b s t i t u t i o n form as a s p e c i a l case i s e l i m i n a t e d . The v a r i a b l e e l a s t i c i t y of s u b s t i t u t i o n f u n c t i o n i n c l u d e s the Harrod—Domar model, t h e l i n e a r p r o d u c t i o n f u n c t i o n and t h e Cobb—Douglas f u n c t i o n as s p e c i a l c a s e s , as does the c o n s t a n t e l a s t i c i t y of s u b s t i t u t i o n f u n c t i o n . The c o n s t a n t e l a s t i c i t y of s u b s t i t u t i o n f u n c t i o n , 6 £ 1 , however a l s o c o n t a i n s the L e o n t i e f t w o — f a c t o r model as a s p e c i a l case which the v a r i a b l e e l a s t i c i t y of s u b s t i t u t i o n model and the Cobb—Douglas model do n o t . The d e r i v a t i o n of the v a r i a b l e e l a s t i c i t y of s u b s t i t u t i o n p r o d u c t i o n f u n c t i o n from the c h o i c e of 6 d e s c r i b e d above, i s not p r e s e n t e d by Revankar i n t h i s paper. I t i s s i m p l y i n t r o d u c e d , and i t s p r o p e r t i e s d e s c r i b e d and c o n s t r a s t e d w i t h the c o n s t a n t e l a s t i c i t y of s u b s t i t u t i o n p r o d u c t i o n f u n c t i o n . I t i s V = A'K | ex Sp 47 where 0 ^ and K / 1-&P It can be seen that the variable e l a s t i c i t y of substitution function has strong a f f i n i t i e s with the Cobb-Douglas function. o<- is a returns to scale parameter, cS is the d i s t r i b u t i o n parameter, and p acts as a substitution parameter, performing much the same function as i t does in the constant e l a s t i c i t y of substitution function, except that i t s range i s always p o s i t i v e , P\ i s of course the scale parameter. When , th i s function reduces d i r e c t l y to the Cobb—Douglas function. The e l a s t i c i t y of substitution for this function i s "Thus the 6 for the VES varies l i n e a r l y with the capital—labour ratio around the intercept of unity. We assume that <J >0 in the empirically relevant range of K / L . This requires that L/K > ( l - p ) / ( l - 6p ) "(Revankar, op.cit.) The VES function s a t i s f i e s the requirements of a neo-classical production function, namely £L - M D , ©V . M n Wr i t i n g i± - L+ (p-l )K , then M - *&p\L > 0 61 ix and $1 -48 Further, since i s the marginal rate of substitution (MRS) of c a p i t a l for land which equals — + 1•T^°• — The MRSRL i s a function of the capital—labour r a t i o , and 6 C ( o V / d K y ( 6 V / 6 L ) l _ n Thus as the ratio increases, the MRSKL decreases. As p increases from p=0 to p c 7<S , the isoquant map fl a t t e n s out. When p = 0, the isoquants are L—shaped ( f i x e d — c o e f f i c i e n t s ) , when p = 1, the Cobb—Douglas f i n c t i o n i s given, and when p ^  V$ > the linear production function r e s u l t s . These are shown in F i g . 1.2.The difference in isoquant maps produced by the VES function as compared to the CES function, as shown for example in F i g . 1.1, should become clear. The isoquants of a CES function are positive segments of hyperbolae with constant e l a s t i c i t y of substitution a l l the way along them. 6 i s constant independent of the le v e l of output and hence at a l l points along an isoquant map. The VES function however, requires only that 6 be the same only along a ray ( OP in Fig.1.2 ), not along an isoquant ( |0 =1 excepted of course ). We now compare the VES function to the CES function and assume oC=1 i . e . , constant returns to scale. The VES function posits the following relations between ^/^ and : V r r r - 1 r - 3-P - r0r+ Kvi Yc ~ i^ sp ' ^ " jL-sp 0 K FIGURE 1.2: Isoquants Of The VES Production Function (After Revankar, 1971) 50 and and X and W given by ^ ( 0 K and ^ / o L - 0 n t n e other hand, i t can be shown that for the CES function the comparable marginal conditions are where Thus the VES function implies that the r e l a t i o n between output and c a p i t a l i s a linear function of the wage rate and the p r o f i t rate, whereas the CES re l a t i o n implies that t h i s i s a log—linear function of the p r o f i t rate. This linear vs. log—linear r e l a t i o n i s even clearer in the case of the cap i t a l — l a n d r a t i o . Income shares are also a linear function in the VES case, and log—linear in the CES case (the log—linear form of the CES i s the only one that reduces d i r e c t l y to the Cobb—Douglas case): HOr)=C*(i-q;)ln(w/r) (CES) As noted the e l a s t i c i t y of substitution in the VES function 51 varies l i n e a r l y with the K/L ra t i o around the intercept term of unity. In fact, when p>1, 6 decreases steadily with t h i s r a t i o and stays below unity over the empirically relevant range of the capital-labour r a t i o , when P<1, 6 increases steadily and stays above unity over the entire range of the capital—labour r a t i o . In other words, in the VES function as written, 6 cannot cross over from <1 to >1, over the empirical range of K/L, regardless of the value of 6. This i s a shortcoming of the VES function, but t h i s crossover can be secured in a time-series context i f the VES function is modified to incorporate nonneutral technical change (in the Hicksian sense, see below). In th i s case the VES function i s given by Vf = ( \ W ( ^ I H ; ( M P ' [ L 1 + ( P - 1 ) ( i - t . ) K 1 ] " 6 ' Neutral technical change i s represented by e.*p(xt) , and Z. i s referred to as the neu t r a l i t y parameter. Non—neutral technical change i s given by (l + Jbt) , and b is the non—neutrality parameter. In the Hicks' d e f i n i t i o n neutral technological change does not a l t e r the marginal productivity r a t i o , whereas a non—neutral technical change i s capital—using, i f i t increases the marginal productivity of c a p i t a l r e l a t i v e to labour, while holding the ratio constant. Empirical investigation using t h i s function indicates that values of 6 tend to be less than unity. A value of 6 <1 means that 6 decreases with the capital—labour r a t i o . This finding implies that as an industry gets more labour—intensive, i t 52 experiences l a r g e r e l a s t i c i t i e s of s u b s t i t u t i o n than when i t gets more c a p i t a l — i n t e n s i v e . 1.3 T e c h n i c a l Change The main i n t e r e s t in the use of these f u n c t i o n s has been in the aggregate growth s t u d i e s . They are meant to provide a framework in which s t a t i s t i c s on growth can be i n t e r p r e t e d , so that i n c r e a s e s i n output/head which are due to t e c h n i c a l p r ogress i . e . , s h i f t s of an aggregate p r o d u c t i o n f u n c t i o n may be d i s t i n g u i s h e d from those which are due to c a p i t a l accumulation or "deepening", i . e . , movements along a given p r o d u c t i o n f u n c t i o n (Hahn and Matthews, 1 9 6 4 ) . The d e f i n i t i o n and measurement of t e c h n i c a l progress i n v o l v e s a p r i o r d e c i s i o n as to which p o i n t on the new p r o d u c t i o n f u n c t i o n should be used to compare with the o l d . The two p r i n c i p a l d e f i n i t i o n s are those of Hicks and Harrod. Hicks keeps the c a p i t a l — l a b o u r r a t i o constant and compares the r a t i o of the marginal products i n the two s i t u a t i o n s . As i n d i c a t e d e a r l i e r , t e c h n i c a l progress i s c a p i t a l — u s i n g i f i t i n c r e a s e s the marginal p r o d u c t i v i t y of c a p i t a l r e l a t i v e to that of l a b o u r . Harrod keeps the marginal p r o d u c t i v i t y of c a p i t a l constant and then looks at what to the o u t p u t — c a p i t a l r a t i o . N e u t r a l t e c h n i c a l progress occurs when the.output—capital r a t i o remains unchanged as a r e s u l t of the 53 change. For the two conditions to be met, the K/L r a t i o must r i s e . Harrod-neutral technical progress thus leaves the d i s t r i b u t i o n of income between labour and c a p i t a l unchanged, assuming perfect competition so that the the vintages of tfoeimarginal products equal factor rewards. The two d e f i n i t i o n s may be i l l u s t r a t e d by reference to fig.1,3. These graphs show the response of output/head (y) to changes in capital/head (k). The l i n e OF in each case-represents the si t u a t i o n before, and OF' the situation after technical change has occurred. In the Hicks' d e f i n i t i o n a l i n e (STT') compares situations where the K/L rat i o s are the same. The slope of OF (OF') at T (T') i . e . , B'T (RT') measures the p r o f i t rate at t h i s point. OW (OW) measures the wage rate in each case. If the slope ,Z:ORW equals the p r o f i t rate (v), and OW the wage rate (w), then, since slope = rise/run = OW/OR, i . e . , r«W/OR, OR=W/r> the wage-profit r a t i o . In F i g . 3a, the l i n e OR gives the wage-profit r a t i o for both OF and OF' when comparison i s made at the point k=s. Technical progress here i s therefore Hicks—neutral. In the Harrod d e f i n i t i o n , the comparison is between points where the p r o f i t rate is the same, i . e . , where the tangents of the slopes of OF and OF' are p a r a l l e l . If these points l i e on a straight l i n e through the o r i g i n , then technical progress i s Harrod-neutral, since the slope of such a l i n e i s the output—capital r a t i o , and i f the l i n e i s straight, this r a t i o does not a l t e r . In f i g . l}.3b, a constant output—capital r a t i o i s compatible with a constant rate of p r o f i t , and technical progress i s therefore 1. Hicks D e f i n i t i o n 55 Harrod-neutral. Note that the K/L r a t i o is higher (h as compared to h' in the situation where the comparison i s made), and that OF' is a r a d i a l projection of OF. If technical progress i s considered a function of time, then the production function may be written \ = f(K, L, t) For the Hicks—neutral s i t u a t i o n , this r e l a t i o n may be expressed as Y t = A(t) f(K, L) Where A(t) is any increasing function of time!! D i f f e r e n t i a t i o n of t h i s function with respect to to any variable leaves the r a t i o of marginal products unchanged. For the Harrod-neutral case, the appropriate r e l a t i o n would be Yt = f (K.Att)U In this case, i t i s apparent that an equal proportionate r i s e in K and A(t)L must lead to an equal proportionate r i s e in Yt . The whole economy r i s e s in scale, and f> remains constant. Harrod-neutral technical progress i s closely akin to population growth. Population growth increases the labour force, L, Harrod-neutral technical progress increases the labour force measured in e f f i c i e n c y units A(t)L. Population growth causes there to be two men where previously there was only one. 56 Harrod-neutral technical progress causes one to be able to do twice what he could have done previously. Technical progress can be both Hicks- and Harrod-neutral. Hicks—neutral technical progress retains the same d i s t r i b u t i o n of income in the -event that K/L remains unchanged. Harrod-neutral technical progress retains the' same d i s t r i b u t i o n in the event that K rises in the same proportion as Y. For both requirements to be met, the d i s t r i b u t i o n of income must therefore be the same as between two situations that d i f f e r only in their l e v e l s of K. In other words, for values of Y/K to remain constant for a given r, and values of w/r to remain constant for a given K/L, then as we move along OF and OF' from comparing Y/K at values of K/L=h and h', to comparing values of w/r at a single value of K/L=S, there must be unit e l a s t i c i t y of substitution of labour for c a p i t a l . Therefore the Cobb—Douglas production function allows for neutral technical progress in both the Hicks and Harrod d e f i n i t i o n s . This gives yet another reason why the Cobb—Douglas production function has been so popular. The Harrod version i s considered to be the more useful d e f i n i t i o n of technical progress, since constant K/L ratios do nt occur very often in the course of technical progress. Technical progress can be either disembodied or embodied. Disembodied technical change aff e c t s a l l machines equally, so the rate of p r o f i t on old machinery i s unaltered by technical progress. With embodied technical progress, the c a p i t a l stock i s 57 altered only via new investment. This reduces the p r o f i t a b i l i t y of the old machinery. For the o v e r a l l rate of p r o f i t to remain unaltered, technical progress must be Harrod-neutral. This d i s t i n c t i o n between disembodied and embodied technical change i s one of the bases of the vintage—non—vintage d i s t i n c t i o n in growth theory. The other i s that, once machinery is i n s t a l l e d , capital/labour ratios cannot a l t e r . (See Hahn and Matthews, 1964, on which th i s section i s based) Vintage models w i l l be discussed l a t e r . We now turn to a review of the use in urban economics of the models thus far discussed. 58 2 NEO-CLASSICAL THEORY IN URBAN ECONOMICS I: UTILITY THEORY APPROACHES U t i l i t y theory i s by far the most common aproach taken to urban economic problems. However, as Richardson (1977) points out, there i s a concealed non—convexity in urban u t i l i t y problems, since a household can be only in one place at any one time. Households have to make an i n d i v i s i b l e l o c a t i o n a l choice and their preferences may vary. It follows that i t i s not optimal for everyone to have the same consumption bundle. Of course, for an optimal solution everyone must have the same marginal u t i l i t y of income. But i f the consumption bundles of households vary, they do not necessarily have the same t o t a l u t i l i t y l e v e l s . In t h i s case, either aggregation of consumers may not be possible, or the problem of choosing between choice sets a r i s e s . The model used here to i l l u s t r a t e the u t i l i t y theory approach i s Beckmann, 1974, "Spatial Equilibrium in the Housing where c i s consumption of the composite commodity, S consumption Market". Assume a log u t i l i t y function, of housing (in square feet), fc i s leis u r e t ime, c o — e f f i c i e n t s of a t t r a c t i o n . 59 The budget constraint V) - P C C + ps(r)S +kr ( where p c , Ps(r) are the prices of C and S, and ki~ i s the cost of travel - actually the amount of time spent t r a v e l l i n g to the CBD - with r being distance ) can be incorporated into the u t i l i t y function, i f the price index for other goods and services i s one as follows: u. - cL 0 l o^rp $>s)-f ajogs + o.j.iogCT-kr) T is disposable time ( t o t a l time minus working hours ), the c o — e f f i c i e n t therefore measures the value of l e i s u r e time. The decision variables of a household are: the distance r at which to locate, and the quantity S of housing to rent ( or buy ). U t i l i t y maximisation requires that Us (^ou/os) and U r vanish a, = - + i t =o )}-Ps s UL — _ a<>s dp_ CL*.k _ Q re—arrangement of terms in the solution for \ i s y i e l d s S = - a ' . 1_ i . e . , housing expenditure is a constant proportion of income. Another re—arrangement yields \| - p, = p S , substitution of this expression into that for U. y i e l d s _ Q-pS d p a^k 60 or p "dr "'diV-ltr This condition contains no variables other than jo and r. It must hold therefore at a l l distances regardless of y and S. The solution of th i s d i f f e r e n t i a l equation i s (03 p =- (ax/O '°<3 ( T - k r ) + logf>0 P(fl ' p 0 ( T - kO putting in the constant of integration, log p o , where r D i s the constant of integration. The gradient of f3(V") i s convex, straight or concave to the o r i g i n f ., depending as \ \ ° - 1 . P0 the i n i t a l condition, i s land rent at the border of the CBD. This i s given by a Cobb—Douglas function k -bx p i . e . , the intensive form with ^ - 6 ^ . The analysis follows the standard neo-classical procedure. P r o f i t i s to be maximised, and the p r o f i t function i s therefore d i f f e r e n t i a t e d with respect to each factor, and set equal to zero, so that the demand expression for each factor can be obtained in terms of i t s price, i t s price e l a s t i c i t y , and revenues, and then substituted into the supply function to give an i n d i r e c t objective function. In th i s case, there i s only one factor, ycL*~}t 1 \ _ t>b Plr) l V(i-p) X C n _ ^ c J where, c b the cost of capital. 61 and f , ( r ) = b ^ l i p ^ f ^ given the expression for p(r) obtained e a r l i e r , p r o f i t per area C\Cr) = t\(0 C%(j)r Substituting for K(r), %(r) and' p(r) gives r^ (o = ( i-c ) ( L ) (bp0) (T-.hr) u p r o f i t / a r e a as a function of distance i s convex, linear or concave depending on whether Next consider the l o c a l equilibrium of supply and demand at any distance r. In a ring of width clr, l e t HOOdf families reside. Their aggregate income i s Vj(r)dr . Aggregate demand for housing then equals ae+ax per) supply i s given by -SCr) dr = ix rh(r)dr Equating supply and demand and solving for N(r} gives substituting for K(r) yiel d s 62 where ) ^ thus substituting for p(r) [ This equation ] says that the equilibrium of supply and demand determines the aggregate income at each distance. It i s , however, completely a r b i t r a r y how t h i s aggregate income i s composed. There i s thus no segregation by family income in t h i s system. Rich or poor do not prefer certain distances over others, but are i n d i f f e r e n t among all o c a t i o n s and hence mix f r e e l y . The basic rent P0 is now determined by aggregating "Y over a l l distances. \ As Beckmann says: A situation where households of a l l incomes are i n d i f f e r e n t among locations is a singular type of equilibrium. Other s p e c i f i c a t i o n s of the u t i l i t y function of the way in which transportation cost enters the u t i l i t y function ( as money rather than time cost, say) w i l l lead to a preferred distance for each income and hence to an income s t r a t i f i c a t i o n by distance. Thus the pattern of r e s i d e n t i a l location in any model depends e n t i r e l y on the c h a r a c t e r i s t i c s of the u t i l i t y function. This i s a consequence of the neo-classical approach which assumes that the system exists to keep the consumer happy, and that supply automatically adapts to meet demand ( e.g., 63 Beckmann, 1973, uses the same procedure to develop a series of models of r e s i d e n t i a l location ). The purpose of using such a function is that the assumption that a l l jobs are located in the CBD may be relaxed: Suppose that there are lo c a l employment opportunities throughout the r e s i d e n t i a l area. If the labour market is competitive, then wages paid in l o c a l employment are less by an amount whose u t i l i t y in consumption equals the u t i l i t y of time saved on t r i p s to work. The equilibrium d i s t r i b u t i o n of housing rents remains unchanged. The model can be used to investigate the effect of taxation on land rents, land values, and the supply of housing. If a fraction t of rents, q,(r) , i s taxed .away, p r o f i t s of landlords are decreased proportionately, but the a l l o c a t i o n decisions remain unchanged. Hence the supply and price of housing remain the same, and the housing market in equilibrium. The net effect i s to transfer income or wealth from landowners to other classes of the urban population. Allocation i s affected d i f f e r e n t l y i f d i f f e r e n t land—uses are taxed d i f f e r e n t l y . (Beckmann, op.cit.) A proportional tax on housing may be considered either in the form of a levy on housing rents, p(r) , or on the value of the property which is the c a p i t a l i s e d housing rent. Let % be the constant tax rate, then landlords seek Ma*{i-t)poobx -ex. giving kr) = \ > * ' - » U \ - - . i l ) M P /CI- P> where pO") i s now 64 and p i s the new rent at X =. T0 . The equation e q u i l i b r a t i n g supply and demand assumes the form a o + a i P 0[T-krJ a i / a i from which ^ = L c i - W f a ' * , - « . c l r . , r l ) assuming that the l i m i t s of the r e s i d e n t i a l area, given by l^ox = f / K , i t follows from a comparison with V= C(r 0 , fj) PQ1^"^ , i . e . , the non-tax sit u a t i o n , that • t ,i The e l a s t i c i t y of rents with respect to to the untaxed proportion (l-?) i s thus . The consumption of every household (assuming no relocation following the tax) decreases by a factor of ' ( l - T f and so does the t o t a l supply of housing. 65 3 NEO-CLASSICAL THEORY IN URBAN ECONOMICS I I : NON-VINTAGE PRODUCTION FUNCTION APPROACHES In dealing with urban growth, labour is normally ignored as an input, and replaced by land in the production function. In a homogeneous p l a i n , the amount of land available i s a simple function of distance ( Tt r ). Distance produces transport costs which must be overcome in order to bring more land into production. Land vintages therefore decrease with distance from the centre, either a market place or a CBD. The production function then describes how t o t a l revenue i s divided between rents and transport costs for any value of K/L, where L now represents land, not labour. The rationale which permits the r e l a b e l l i n g of the symbol 'L' in the production function i s that land, l i k e labour i s a non—produced factor of production. C l a s s i c a l economists such as Ricardo and von Thunen created models in which land and labour, or labour with c a p i t a l , were the only two factors of production*. 66 The wage paid to labour i s an h i s t o r i c a l l y determined subsistence minimum, and rent accrues to the landlords as a residual. The more productive the land, the greater the residual accruing to the landlords. Landlords could claim therefore that (variable) rents to land were due to i t s (variable) productivity alone, that labour was uniformly productive and received a uniform wage, and that when necessary, the owner of machinery received a p r o f i t calculated at the 'normal' rate. To j u s t i f y t h i s payment of p r o f i t to the owner of c a p i t a l involved a re—formulation of the theory in terms of marginal products. This was achieved by Wicksteed, who f i r s t stated that i f input and output markets are competitive so that land rentals and labour costs both equalled the value of their respective marginal products, ( i . e . , the value of the extra output produced by the last parcel of land or the l a s t worker that could be employed consistent with a positive rate of p r o f i t — the 'extensive' and 'intensive' respective margins of production), then, i f constant returns to scale apply, the application of Euler's theorem 1 in a corn economy, with c i r c u l a t i n g c a p i t a l only, corn serves as the 'wage fund', to feed the labour—force over the year which elapses between seed—time and harvest. The amount of capital(corn)/worker i s thus equal to the annual wage/worker. Ricardo thus treats labour—with—capital as a single input (cf. Sraffa, 1951, Howard and King, 1975). 67 meant that, faute de mieux, the remainder must i equal the value of the marginal productivity of c a p i t a l ( i . e . , the marginal productivity of c a p i t a l multiplied by the product price set by the market; see also Dobb, 1940). Thus, where c l a s s i c a l theory adopted a factor cost adding—up procedure to arrive at a market price, neo-classical production function theory, in i t s micro—economic applications, takes the market price as given and divides the revenue on the basis of a t a u t o l o g i c a l l y determined assessment of contribution to productivity. In i t s macro—economic application the value of t o t a l output i s always assumed s u f f i c i e n t l y large to s a t i s f y the consumption requirements of a l l s o c i a l classes. The conditions set down by Wicksteed are pQt = p-MP^L + p-MVN + P'MIVK As stated e a r l i e r , under competitive conditions, the value of the marginal products are equal to their respective prices, so that this expression may be written The sum of the competitive payments to the three factors or production i s exactly equal to t o t a l receipts from the sale of the product on a competitive market. This remarkable result i s a basic r e c o n c i l i a t i o n of the c l a s s i c a l theories of land rent. ( M i l l s , 1972b) Not a l l would concur with M i l l s ' assessment however. Garegnani, (1970) reviews these developments as follows: The way was open, i t was thought, to explaining in terms of marginal productivity, the d i v i s i o n of the product between labour and c a p i t a l , which the c l a s s i c a l economists had analysed by alto—gether 68 d i f f e r e n t p r i n c i p l e s . But in order to explain the rate of p r o f i t along these new l i n e s , c a p i t a l had to be conceived ultimately as a single magnitude: and had accordingly to be measured as a value quantity; unlike labour or land which were physical quantities. The extension of the "law of rent" to d i s t r i b u t i o n between labour and c a p i t a l therefore raised the danger of c i r c u l a r reasoning: the value of a c a p i t a l good, l i k e that of any other product, changes with those very rates of wages and interest which are to be explained by means of the "quantities of c a p i t a l . " The marginal parcel of land or the marginal worker i s a physical quantity, the value of which can be estimated from i t s p r i c e . In equilibrium, marginal cost (MC) equals average cost(AC). For land and labour, the measurement of average cost by d i v i d i n g the rental or the wage b i l l by the t o t a l physical quantities of land or labour employed i s technically possible (Keynes, 1936). The aggregation of c a p i t a l into a single index for the purposes of estimating i t s average or marginal cost depends on being able to value i t by an index which does not vary with variations in r e l a t i v e prices or d i s t r i b u t i o n , since the comparison of two tons of p r i n t i n g presses with two tons of t e l e v i s i o n sets i s a meaningless one. We s h a l l return to these issues l a t e r , but note for the time being that the treatment of land in neo-classical urban economies as a mere c o r o l l a r y to the use of labour in the neo-classical production function is not without irony since the neo-classical approach purports to be a re—formulation of the c l a s s i c a l interpretation of the d i v i s i o n of income between labour or labour—with—capital and land. To quote from Richardson(1977, p. 25) NUE [ New Urban Economics ] models do not provide 69 an o v e r a l l view on the nature of land r e n t . An obvious support of t h i s statement i s the f a i l u r e of NUE to say anything about n o n — r e s i d e n t i a l rent (almost always t r e a t e d as exogeneous), or a f o r t i o r i , about i n d u s t r i a l r e n t . Less apparent, though e q u a l l y s i g n i f i c a n t i s that NUE t h e o r i e s , and the Alonso—Wingo—Muth h e r i t a g e from which they evolved, l i e o u t s i d e the mainstream of land rent theory... Land i s not valued f o r i t s own sake.... but because of i t s l o c a t i o n . High rents are a surrogate f o r the b e n e f i t s of a c c e s s i b i l i t y . I t i s f o r t h i s reason that NUE modellers have focussed on r e s i d e n t i a l space per se, rather than on housing p r o d u c t i o n , a s e c t o r that uses land as an important input (Borukhov, 1973, M i l l s , 1972a are rare exceptions to t h i s g e n e r a l i s a t i o n ) . 3.1 Cobb—Douglas Approaches Without f u r t h e r ado, t h e r e f o r e , we turn to a c o n s i d e r a t i o n of M i l l s ' approach, both i n i t s own r i g h t , and as a example of the use of the Cobb-Douglas p r o d u c t i o n f u n c t i o n . The extended v e r s i o n of t h i s model i s contained i n M i l l s ( 1 9 7 2 a ) , the simpler v e r s i o n i n M i l l s (1972b). As the extended v e r s i o n of t h i s model comp l i c a t e s the method without a l t e r i n g the substance s i g n i f i c a n t l y , we s h a l l o u t l i n e the steps as presented in M i l l s , 1972b. A Cobb—Douglas pr o d u c t i o n f u n c t i o n , (where X s(iO i s output of housing s e r v i c e s at d i s t a n c e u from the c i t y c e n t r e , and L(oC) and are the land and c a p i t a l inputs r e s p e c t i v e l y ) g i v e s the supply, and 70 ( where X^CuO i s the demand/worker l i v i n g at ix for housing services, ' B i s a scale parameter, w i s wages, p(u) i s the price of housing services at u, and Q A and © x are the price and income e l a s t i c i t i e s , respectively) gives the supply. Aggregate demand at a i s (where N($ i s the workforce l i v i n g at a distance u from the c i t y centre ), and in equilibrium X = X * t<0 The problem i s therefore to solve t h i s r e l a t i o n for land rentals, "^.(^ , in terms of econometrically estimable parameters, namely oc., QX , \ , H and tr (the rate of p r o f i t is assumed to be exogeneous to the urban economy, and so is a given) given households' r e s i d e n t i a l location requirements, namely p ' C u ^ x ^ + t c 0 where jp'OA i s the slope of p(W.\ and t i s commuting cost. The r e l a t i o n states that households are unable to increase u t i l i t y by moving, i f the change in the cost,of housing i s just offset by the change in the commuting cost. Other conditions are that no land be unavailable for housing, that the rent at edge of the c i t y be set by non—urban land rents, i . e . , exogeneously, and that a l l the land available be occupied. These three conditions are that the amount of land at UL, L(jCi be ( where ^ i s radians - M i l l s therefore assumes that the c i t y i s 71 semicircular); that the rent at edge of the c i t y \A i s and that the t o t a l numbers of workers in the urban area i s U where i s the edge of the CBD. The procedure used is to p a r t i a l l y d i f f e r e n t i a t e the housing supply function with respect to land and c a p i t a l , re—arrange the r e s u l t i n g expressions to obtain demand expressions for land and c a p i t a l in terms of their prices, the product price, and the c r o s s - e l a s t i c i t i e s of factor demand (c< and \-°^ ) and substitute these back into the housing supply function, which in t h i s guise i s then the i n d i r e c t objective function, only of revenues, factor prices, and their e l a s t i c i t i e s . Re—arranging terms gives Note that this procedure r e l i e s on the assumptions that **>hp(uO - "ROJO and • D C a ) = r . T h e S e two 6 Lea) r dK ( l O ) 1 assumptions are v i t a l to neo-classical procedures. This expression for p(u^ can then be substituted into the expression for housing demand/worker, and the resulting expression into the expression for l o c a t i o n a l equilibrium. D i f f e r e n t i a t i n g p(U-) and substituting t h i s into the equilibrium condition, completes the transformation of t h i s expression into the i n d i r e c t objective function. It is composed of constants and only. It i s 72 (where E and p stand for c o l l e c t i o n s of constants: and p = oc(i+eJ ) # Solving this d i f f e r e n t i a l expression in terms of and using the i n i t i a l condition , the solution i s M i l l s refers to evidence that the price e l a s t i c i t y of housing demand i s in fact close to -1 and therefore the value of (3 to use i s zero so that l^ uY* RexpCtEca-u^), i . e . , a negative exponential f unct ion ( U - ). The r e s i d e n t i a l density, NM|L(u\ can be obtained from the expression N(^=)Cs(u^ f i f the same sequence of substitutions i s made for TL^) , and the following sequence of substitutions made for )Cs(uO and thus The expression for housing following a l l these substitutions is The result of these substitutions i s that L c« 73 (and after some re—arrangement The expression E is the reciprocal of the labour force p a r t i c i p a t i o n rate, the expression 0~fO i s approximately one, so that the exponential version of l^ w) applies, and Population density i s proportionate to land rent and declines with distance as does land rent. M i l l s tests t h i s model using estimates of the relevant parameters. 3.2 Constant E l a s t i c i t y Of Substitution This model of M i l l s , though fa m i l i a r , has been given a f a i r l y thorough treatment, for the purposes of contrasting i t with the constant e l a s t i c i t y of substitution production function as applied by Kau and Lee (1976) to the urban case. The exposition i s aided by the fact that Kau and Lee couch their discussion in terms of the M i l l s model. Supply i s given by Demand/worker is as before. Rental and p r o f i t rates are given by the value of the marginal products as before 74 Solving these equations for L ( L O and and substituting t h i s into the housing supply function gives Compare th i s with the Cobb—Douglas case port = HA*-u+*V** -]: 1r^Rcu)*' which is m u l t i p l i c a t i v e rather than additive. The price e l a s t i c i t y of housing with respect to r and is 6Rc^ p(a^  L A- 1 ROM 4 The CES function allows the e l a s t i c i t i e s of substitution to vary with changes in r and *R(u). These e l a s t i c i t i e s w i l l reduce to I - < * and o t when p =1, i . e . , Cobb-Douglas; in the case of fixed c o - e f f i c i e n t s they w i l l become A'V/pOx) and K% R(U)/p(tt) respect i v e l y . Again the equilibrium condition is p'uox3,(io +e =o or pV)Bw9' p(u)\t =0 75 Using the i n i t i a l condition of equation 3 . 2.1 the solution to this d i f f e r e n t i a l equation can be written where ROJO = ^ ' V < : i + P A P / ( 1 + P ) [p^ + 1 + ^ a m + i ) J/VK+»C*+P«_ ^ A ^ y l L ^ 0 + p > this function has three l i m i t i n g forms, when Qx~* ~ , when p_» 0 , and when 6x-»-l and p -* 0 When ^-» r l , the rental formula becomes •R ^ ={(«) n » t c a . ^ *p)/p i . e . , land rent i s inversely related to distance, and the rental rate of c a p i t a l . When P Q , we have (where p = [fWkC>-*l""*i J ' 1 ^ ^ ) i . e . , land rent i s inversely related to distance. This reduces to the M i l l s formula when j(? £ 0 When 6X=-1 and p ->0 , we have i. e . , the case generally assumed by M i l l s . The re l a t i o n between population density and land rentals i s obtained in the same way as M i l l s , i . e . , The expression for K (a) , given by . taking the r a t i o of the 76 marginal products i s Demand/worker, obtained in the same way as before, i s Density, i s then and i s a positive function of and decreases with increasing distance and r. Again, there are three l i m i t i n g cases. As G ^ - l t h i s again i s a positive function of and decreases with increasing distance and f . As D 0 This reduces to M i l l s formula when upon making the appropriate substitution for "^C^ . As p-^Q and 9 X " 1 This i s M i l l s formula when (9-0. The biases associated with choosing 6 1 = - 1 , p = 0 , can be estimated in the following way. The f i r s t case is the bias associated from assuming 0 =-}. Assume P =0 for exposition's 77 sake. Again a form of similar to that of M i l l s can be written as: where as before, and v E ^ / I f t A i ^ ^ V f a l s o a s b e f o r e . The econometric technique for estimating in the case of is are re(\r<?ssior\ for >^~Q, i t i s ' f ' i f (X>W - C / X = - H II \ ,V ate arc ^ ^ . i e n t e A i s known as the functional form parameter. This expression may be transformed into ^ ioo>x^  = X - ^ * - &D<$ft&] where £> - ^A., A<1, and ^ = unbiased density gradient. This implies that the density gradient obtained by assuming O^-"! w i l l be biased by a factor - 00^4^3^ unless % i s not s i g n i f i c a n t l y d i f f e r e n t from zero. Dropping the assumption that 0 gives: " ^ ( U L ^ ^ H R ^ + ^ ^ - U ^ + » o m i t t e d terms associated with the e l a s t i c i t y of substitution" (sic) (Kau and Lee op.cit.) Kau and Lee conclude that: In sum the density gradient obtained by is subject to two possible biases: ( 1 ) the bias associated with the e l a s t i c i t y of substitution parameter and ( 2 ) the bias associated with the price e l a s t i c i t y . The density gradient obtained from [(TXuy^jyx - - ] i s subject only to one 78 possible bias, that associated with p . . . . ...A completely s p e c i f i e d density function can be obtained...A d i r e c t method of estimating the density gradient for this complete form is s t i l l an open question. However i f the e l a s t i c i t y of substitution parameter , %f>, and the price e l a s t i c i t y , Q% , are estimated independently, then these estimated values could be used in the complete form along with numerical analysis techniques to estimate the density gradient. This not perhaps a very s a t i s f a c t o r y conclusion. The same forms of the density gradient that are empirically estimable ^(derived by M i l l s using the far simpler Cobb-Douglas form. "This indicates a basic d i f f i c u l t y with neo-classical approaches; once the very simple assumptions are removed, the models quickly become empirically, not to mention mathematically, intractable (Kaldor, 1972). However M i l l s was only able to estimate densities on the assumption that ^ = " " 1 . This led him into some inconsistencies, for example, in his extended model, in his solution for the price of housing, 0 A has to be set at -1.5, or the formula becomes unstable. Kau and Lee are able to show what the s p e c i f i c a t i o n error w i l l be, and how to estimate density on the assumption that |3 fO. But t h i s has l i t t l e to do with use of a constant e l a s t i c i t y of substitution function. A more interesting aspect of t h i s work i s that they have one of the few discussions of the analogies between technical change and urban growth, that are i m p l i c i t in the use of production functions in urban contexts. Unfortunately, their discussion i s in terms of Hicks d e f i n i t i o n of technical progress, when have seen that Harrod's d e f i n i t i o n i s regarded as 79 being the more useful. However they refer to neither author by name: If the marginal product of c a p i t a l rises r e l a t i v e to the marginal product of land... there i s a land saving technological change. As shown in previous studies, when c a p i t a l i s growing faster r e l a t i v e to land, technological progress, which eases the substitution of c a p i t a l for land, w i l l be c a p i t a l using. (Kau and Lee op.cit.) However, especially since the production function employed is not Cobb—Douglas, such technological change could be also Harrod neutral, or even Harrod land using, depending on by just how much c a p i t a l i s growing faster than land in the composition of f i n a l output. A consequence of the Hicks d e f i n i t i o n i s that attention i s focussed on the marginal products. Kau and Lee use a r e s u l t , taken from Brown 1968, that when <5 increases, "output w i l l increase for a given set of factors of production. There is thus a positive r e l a t i o n between the e l a s t i c i t y of substitution and Lee, 1976) Bearing in mind Garegnani's comments, noted above, i t i s d i f f i c u l t to see how factors of production can be regarded as "given" in the face of changing e l a s t i c i t i e s of substitution, or a f o r t i o r i in the face of technological progress. The remainder of the discussion i s s i m i l a r i l y confusing. F i r s t , the sign of the supply of housing services, i.e (Kau and 1 s indeterminate, though by the application of 1 Hopital's rule, the signs of the l i m i t s are 80 and T h i s i n d i c a t e s no r e l a t i o n between the p r i c e of h o u s i n g s e r v i c e s and the e l a s t i c i t y of s u b s t i t u t i o n , nor by e x t e n s i o n , between the demand f o r hous i n g and 6 i . e . , &*if u Vop = 0 . can be shown t o be i f l o c a t i o n a l c h o i c e i s assumed t o be independent of the demand f o r h o u s i n g s e r v i c e s as i m p l i e d by =0, i f c a p i t a l i s growing f a s t e r than l a n d then ^&(op >0. F i n a l l y , the r e l a t i o n between p o p u l a t i o n d e n s i t y and 6 i s a n a l y s e d by assuming t h a t t o t a l p o p u l a t i o n , (note t h a t everyone's h o u s i n g r e q u i r e m e n t s a r e s a t i s f i e d , and so the u t i l i t y m a x i m i s i n g s u p p l y of h o u s i n g s e r v i c e s i s d i v i d e d by the u t i l i t y maximised demand/worker = XT>(u} /""N'C^V r l j p - X j l r t •• " £ ^ = N W ). This way of estimating U^ A depends c r u c i a l l y on the assumption t h a t u t i l i t y i s s a t i s f i e d . The d e r i v a t i v e of NKj*} w i t h r e s p e c t t o p can be dete r m i n e d by the s i g n s of cV^ /oP and C^ Xsl&P* ^^J^p =0. From Brown(1968) , i s n e g a t i v e . T h e r e f o r e 6N(£|6p i s n e g a t i v e . An i n c r e a s e i n 6 l e a d s t o an i n c r e a s e i n N , and a l s o i n d e n s i t y , though i t has no e f f e c t on h o u s i n g demand. In f a c t , of c o u r s e , i t does have an e f f e c t s i n c e s u p p l y i s assumed t o e q u a l demand. However, b a c k t r a c k i n g a l i t t l e , examine the c o n d i t i o n s under which was o b t a i n e d . To o b t a i n the p a r t i a l 81 der ivat ive of with respect to to 0 , n and L must be held constant. However, since the cause of changes in p i s that of technical change, the assumption that K can vbe held constant while d i f f e r e n t i a t i n g with respect to p i s contradictory. Kau and Lee then discuss the implications of r i s i n g 6 for urban growth. They argue that r e s t r i c t i n g the supply of land while technical progress i s increasing the s u b s t i t u t a b i l i t y of c a p i t a l for land would lead to higher densities in the course of urban growth. "This may result in more congestion, higher rents and more p o l l u t i o n . Thus i f the desired goal i s a better urban environment, p o l i c i e s which r e s t r i c t the available land supply may not be successful." This i s of course a non sequitur. Remember that the population estimate was obtained from a consideration of the amount of housing services/head, as determined by the demand for housing services/head. Whatever the population may be therefore, i t s housing requirements are met and i t s u t i l i t y maximised, whatever the density of development. The e x t e r n a l i t i e s , i f there any, and whether or not they may be negative,are part of the deal. The basis of t h i s discussion, that technical progress i s measured by an increase in the e l a s t i c i t y of substitution highlights a common f a i l i n g in neo-classical economics, the confusion of a process with a s i t u a t i o n . If land or labour are r i s i n g r e l a t i v e to c a p i t a l costs, then, under certain limited circumstances, a technical progress w i l l be accompanied by the substitution of c a p i t a l for labour, and i f technical progress 82 continues, t h i s substitution of c a p i t a l for labour may continue at an increasing rate. At any instant however, the e l a s t i c i t y of substitution between the then given c a p i t a l stock and the labour force may have any value from zero to i n f i n i t y . If i t were i n f i n i t e however, the impetus to continued technical progress would be lost since any changes in factor costs could be absorbed by substitution. An e l a s t i c i t y of substitution of zero would be more l i k e l y to stimulate technical progress. Thus over time, the effect of technical progress, cet. par. w i l l be to create e l a s t i c i t y of substitution, but t h i s i s a c h a r a c t e r i s t i c of the technical progress i t s e l f , not of c a p i t a l or labour in production at any instant. The importance of Kau and Lee's work i s therefore li m i t e d . Their most important finding i s that unless 0^ =0 and Q^=-\, the negative exponential r e l a t i o n between density, rent and distance does not hold. 83 3.3 Variable E l a s t i c i t y Of Substitution Sirmans, Kau and Lee (1979) have investigated the use of a VES production function. They begin by investigating the effect of c5 on the intensity of r e s i d e n t i a l land—use in terms of the value of housing produced/unit of land. If i t i s assumed that house prices and wages vary only with distance from the CBD and that there are similar production functions for each unique location, and that producers of housing combine land and c a p i t a l in production to maximise p r o f i t s so that IT , p ^ K j Q t L . K W L - c K (where "JT 1 S p r o f i t s , p i s price of housing, dependent on |_ and Y\ , and Q i s output, with r, C r the price/unit of land and c a p i t a l , respectively) Then pQ/l_ ^ s the intensity of r e s i d e n t i a l land use discussed above. It i s assumed that d i s a function of distance, and therefore has a varying e f f e c t on the slope.of pQfL* This effect i s s p e c i f i e d by the second p a r t i a l derivative of pQJ[_ with respect to distance, where R _ , R - - ^ = -The f i r s t term contains -1^ ^ 0, Muthd 969). The second 84 term represents the bias associated with assuming a Cobb—Douglas function and is 4 0 The t h i r d term represents the bias associated with assuming constant e l a s t i c i t y of substitution, and provides the impact of changing e l a s t i c i t y on density. It also i s ^ 0. The second and t h i r d terms both indicate a density function concave to the o r i g i n , whereas the f i r s t term would indicate a convex curve. The negative curvature of the second two terms tends to offset the positive curvature imparted by the log—distance function. Hence the density gradient may be approximately constant, even though the price gradient declines with distance from the CBD. If the impact of variations in the e l a s t i c i t y of substitution i s greater than the convexity imposed by the price density function, then the density gradient may have a negative curvature, so that population densities would decline less rapidly than negative exponentially with distance from the CBD. In thi s case, i f the price gradient i s held constant, an increase in demand for housing would increase the output of housing and population more rapidly in the outer part of the c i t y . (Sirmans, Kau and Lee, 1979) The VES formula used by Sirmans, Kau and Lee i s the one that allows for technical progress, namely: where H*(^ i s housing services output at U. , and E l a s t i c i t y of substitution i s rs - d , (p-aa+frQi tun} i - s p J i_toa Demand for housing services at U. at time t is e. Marginal products are 85 and The marginal rate of substitution of c a p i t a l for labour i s 6 L * ( « * - > ' S p S p K*CU> Assuming technical change is Hicks c a p i t a l using, the'time rate of and i t s e l a s t i c i t y are derived. The rationale once again being that "higher K t ^ j j J ^ r a t i o s lead to a greater rate of capital—using inventions." Fortunately, t h i s l i n e of reasoning is not pursued any further. The expressions for the marginal products can be re—arranged to give the demand for L^ClA and ^*(>) and K t « A - - - P t ^ ^ v C t , r ^ P , l X l , 6 0 Assuming that oC=i (constant returns to scale), the expression for the price of housing, pt(u) r may n°w be given: The price e l a s t i c i t i e s of housing with respect to and V t ( l l \ are and ' fc 86 These reduce to j.-£> and <£> when jO =1. Household equilibrium i s given once more by Pfc V ) C^ CtO + T = 0 (r = uW a»t } dfr(u) can be replaced in this expression to give This d i f f e r e n t i a l equation may be solved to give an expression for rentals: . "This expression implies that the e x p l i c i t r e l a t i o n between ^(LO) and II cannot be derived unless p =1" (Sirmans, Kau and Lee, 1979). As p =1 gives a Cobb—Douglas form i t i s not surprising that t h i s expression reduces to the one given by M i l l s except for the time variable (expxt ). The r e l a t i o n between population density is derived as before An expression for involving L-ttP) i s obtained from the r a t i o of the expressions for and U($ in terms of the marginal product of \\$), t h i s epresssion for i s substituted into the expression for H*C"5, dtOfy i s given by p^"Bwdl^(U)®x , and density i s 87 "this i s the population density in terms of C^. t t , and other parameters" (Sirmans, Kau and Lee). This reduces to the M i l l s version when p = 1 i f time is removed. (unfortunately, to obtain an e x p l i c i t form for the density, P must equal 1, though thi s may not be so important i f numerical estimates may be obtained.) The p r i n c i p a l result is that i f p fO, then c a p i t a l costs and land rents are required to explain the change of population density in a c i t y , since the VES function takes the K / [_ r a t i o e x p l i c i t l y into account. Compared to the Cobb—Douglas and CES versions, i t produces a density gradient function which decreases at a numerically faster rate with distance. (Sirmans, Kau and Lee, 1979) Whilst Sirmans, Kau and Lee are able to provide empirical estimates of the value of 6 for the VES function, (0.090 -0.929; 0.514 at the mean ^/L r a t i o ) , and have th i s in the predicted r e l a t i o n with the r a t i o , (which decreases from 1.336 - 0.104), and contrast t h i s with the CES function, ( 6=0.619 for the same data), and demonstrate the effects of bias involved in assuming a Cobb—Douglas function, i t i s doubtful whether t h i s work does represent the way forward. Again, as in the CES case, the model quickly becomes unwieldy, in c r u c i a l instances one must rely upon the assumption that p = 1 f ^ u t there is another problem at issue. The f l e x i b i l i t y of the VES model i s also a drawback. While i t is true that varying estimates of 6 88 can be obtained from a VES function without any parameters being affected, since i t s range depends on the r a t i o of the variables, K , and L , i t is d i f f i c u l t to conceive of a production process which would have th i s quality, especially since on Kau and Lee's (1976) and Sirmans, Kau and Lee's own view that the value of 6 i s such an important parameter in production technology. A variable 6 may allow for better "goodness of f i t " of the VES model to any given data, the purpose for which i t was developed, but i t cannot add too much in the way of analysis. The issue i s reminiscent of F. M. Fisher's conclusion as to the e f f i c i e n c y of the Cobb—Douglas function in explaining growth and d i s t r i b u t i o n in U. S. Economic history. The reason that i t gives such results i s that factor shares are r e l a t i v e l y constant, not because the underlying aggregate production technology is Cobb—Douglas: If this i s a useful observation about the real world, then the question of what keeps factor shares r e l a t i v e l y constant i s an open one of obvious importance. (F. M. Fisher, 1971) 89 3.4 Non—Vintage Production Function Models: Conclusion To conclude this section, l e t us remind ourselves of the object of these exercises. It i s to provide estimates of the variations in land rents and c a p i t a l / l a n d r a t i o s , given c a p i t a l costs, and the pattern of d i s t r i b u t i o n of income between land and c a p i t a l , e l a s t i c i t y of substitution, returns to scale, the price and income e l a s t i c i t i e s of demand and i m p l i c i t l y transport costs costs. Transport costs costs reduce the attractiveness of a p a r t i c u l a r piece of land, from either the production or consumption point of view, since land in the urban economy i s not viewed as being productive in i t s e l f , but only as a location for the production of either physical goods or housing services. Land rent therefore r e f l e c t s the productivity, not of land, but of c a p i t a l (the theory of economic rent). The rate at which land rent decreases away from the urban centre is modified by K j\ substitution, and the factors ultimately a f f e c t i n g that, but the cause of i t s decrease i s ultimately transport costs. However sp e c i f i c a t i o n of the factors that modify transport costs in these models depend on there being an invariant value magnitude which can be assigned to the c a p i t a l component of the production function so that i t may be held constant for the purposes of p a r t i a l d i f f e r e n t i a t i o n . As w i l l be shown l a t e r , t h i s requirement i s the cause of the prediction that given two economic actors (usually, but not necessarily, households), d i f f e r i n g only by income, that the poorer one, for whom transport costs assume a greater burden, w i l l always be located closer to the centre, despite land costs being higher there. 91 4 PRODUCTION FUNCTION VINTAGE APPROACHES To date, in th i s review of the l i t e r a t u r e , a number of nagging problems have continued to appear. These have included the e f f e c t s of technical change on the value of the c a p i t a l stock ( discussed in rel a t i o n to the CES function), the automatic response of housing supply to consumers demands ( discussed in re l a t i o n to Beckmann's model ), and the unimodal s t r a t i f i c a t i o n of income groups by distance from the CBD, whenever transport cost i s , introduced into the budget constraint. However, a l l these questions derive ultimately from the problem of having to put a value on c a p i t a l in order to measure i t . The issue of the comparison of pri n t i n g presses and T.V. sets was used to i l l u s t r a t e t h i s . However the point s t i l l stands whether the comparison i s of printing presses of di f f e r e n t technical s p e c i f i c a t i o n s , or of d i f f e r e n t ages only, or the same press under di f f e r e n t circumstances, for example, di f f e r e n t rates of p r o f i t . The use of aggregate production functions, in which c a p i t a l i s continuously substitutable for labour, abolishes the problem by assuming that a l l c a p i t a l goods are made out of the same substance which can be instantaneously and c o s t l e s s l y transformed from one form to another, and even into consumption goods. In such circumstances, a unit of c a p i t a l 92 always retains i t s value. This might be thought of as the 'stock market' d e f i n i t i o n of c a p i t a l , in that shares y i e l d returns and money can be shifted from one investment to another rapidly and ( r e l a t i v e l y ) c o s t l e s s l y . However, the act of investment in shares i s not the same as the act of investment in machinery. The investor in shares in fact assumes part of the debt incurred by the physical investment undertaken by any firm. As Robinson (1953) stated, c a p i t a l i s either an as yet uninvested sum of money earning no rewards, but capable of being put into any form of production, or i t i s machinery in which the owner of c a p i t a l has invested, and to whom i s due a share of the output. Capital i s either uninvested and f l e x i b l e , or t i e d up in machinery and earning a p r o f i t . It cannot be both at the same time. It i s because c a p i t a l is conceived in thi s 'stock market'fashion that supply constraints are non-existent and the pattern . of re s i d e n t i a l location e n t i r e l y a result of the arguments of the u t i l i t y function. If l o c a t i o n a l patterns are derived from the rent gradient given by the production functions employing these concepts of c a p i t a l , the requirement that the marginal products give the values of the returns to land and c a p i t a l means that monotonic decreasing rent gradients are an inevitable consequence of the form of these functions. The very existence of a marginal product of c a p i t a l stemming from such a function i s e n t i r e l y dependent on the use of a concept of c a p i t a l , the value of which can be held constant for the purposes of d i f f e r e n t a t i o n . Thus 93 these monotonic decreasing rent gradients are a result of the use of these homogeneous c a p i t a l concepts, whether one begins with u t i l i t y or production theory. The neo-classical answer to t h i s problem i s the vintage theory. The essence of the approach i s to aggregate production functions (which form vintages) rather than to aggregate c a p i t a l stocks to put in a single aggregate production function. The production function gives a set of potential combinations of labour and c a p i t a l which produce a given output. However, once the p a r t i c u l a r combination of labour and c a p i t a l has been chosen and invested i n , that combination i s fixed. If technical progress occurs, i t can only be introduced to the c a p i t a l stock via new investment. If d e t e r i o r i a t i o n occurs in the c a p i t a l stock so that output is reduced, labour cannot be shed. This type of model is known as 'putty—clay' ( i . e . , varying c o — e f f i c i e n t s of production ex ante, fixed c o — e f f i c i e n t s ex post), i f some technical progress or adjustment to deterioration i s allowed to occur after investment has taken place, a 'putty—putty' model r e s u l t s . However, though these are important differences, since the vintages are aggregations of production functions, i t i s not surprising that vintage models can be made to y i e l d results similar to non—vintage models. Harrod-neutral technical progress i s assumed to be occuring so that output/head increases at a steady rate, but the rate of p r o f i t on c a p i t a l remains constant as do the capital/output r a t i o and the r e l a t i v e shares of wages and p r o f i t s . A constant share of gross investment in t o t a l output then produces growth of output/head at a steady rate. . . The level of wages in terms of product r i s e s in step with output/head (this follows from the condition that the rate of p r o f i t and the share of wages in output are constant) and the equipment for each technique i s scrapped when wage absorbs i t s whole output so that i t s quasi—rent i s reduced to zero. A higher share of p r o f i t e n t a i l s a wider gap between the wage rate and output/head with the l a t e s t , best technique. Thus i t e n t a i l s a longer service l i f e of equipment. Therefore a higher proportion of older, more i n f e r i o r , techniques in use at any given moment, and lower average output/head" (Robinson, 1970) Machines thus have f i n i t e l i v e s for economic reasons, not because of physical collapse. The length of l i f e of machines [written T] is a variable that adds a new dimension to the model as compared with non—vintage models. A condition of steady state growth is p l a i n l y that T should be constant over time.... Because c a p i t a l i s non— malleable, an act of of investment commits a hostage to fortune... Expectations determine the choice of technique at a given wage and the valuation of the c a p i t a l stock. They therefore determine the path followed by the economic from any starting point. Most vintage models assume that income i s expected to grow at the steady—growth rate [bringing in perfect information and the r e a l i s a t i o n of expectations by the back door]... The existence of the length of l i f e of equipment as an economic variable adds an "extensive" margin to whatever intensive margin there may be in the substitution of labour and c a p i t a l . Productivity per man depends on (a) the c a p i t a l intensity of the machines in use;(b) their average age. Equilibrium conditions must be s a t i s f i e d at both margins. There is an obvious a f f i n i t y to the Ricardian theory of rent. Different vintage machines are comparable to Ricardo's acres of d i f f e r i n g f e r t i l i t y . The machine on the margin of being scrapped earns no quasi—rent and corresponds to Ricardo's no rent land. Wage—costs on the marginal machine absorb the whole value of the product. The wage of labour i s therefore equal to the average product of labour on the oldest machine in use. This can be regarded as the marginal product of labour, since i f the supply of labour were s l i g h t l y 95 reduced, competition for labour would drive up wages and make the marginal plant unprofitable. The marginal plant would be scrapped, and the output that i t formerly produced would constitute the net reduction in output in the economic as a whole due to the reduction in the supply of labour... With constant population the labour needed to man the machines that are currently being built, in each period i s made available by the scrapping of machines T years old. (Machines of intermediate age do not surrender any of their labour force, despite the r i s e in wages, because of the absence of ex post substitution). A consequence of this i s that under conditions of steady growth, with constant population, the labour force must be divided equally between the di f f e r e n t vintages. The more modern machines have lower labour requirements, because they embody technical progress, but they are more numerous, because they are the f r u i t of saving out of a higher income. (Hahn & Matthews, 1964) It is easy to see the attractiveness of such a model for urban economics esp e c i a l l y when the Ricardian analogy i s drawn so e x p l i c i t l y by Hahn and Matthews, even i f i t does seem to be a long way round to reach t h i s p o s i t i o n . Two authors who have been so attracted are Muth(l973, 1975) and Evans (1975). Evans' model i s a putty—clay one and so offers a more dire c t comparison to the economic growth model described above. Muth's i s a putty—putty type and w i l l be considered l a t e r . In Evans' model ground rent charged by landlords are analogous to wage rates. Population is assumed to be growing over time in a fixed area. The population has zero price e l a s t i c i t y of demand (though t h i s i s not spelt out by Evans), and so increasing population results in demands for units of the same quality , but at higher density. Ground rents are increasing continuously, because the landlords charge the rent they would 96 earn i f the ex i s t i n g buildings, whose density i s fixed, were meeting present density requirements. Developers thus earn a decreasing quasi—rent on their buildings, and face two decisions, the c a p i t a l intensity, or density of development, and the length of l i f e to hold i t in the housing stock. These are fixed simultaneously by the rate of increase of ground rentals and the r a t i o of ground rentals to developers' returns when new. The proportion of ground rentals to developers' returns on new buildings w i l l vary as between situations of zero and perfect foresight. If ground rentals are not expected to r i s e (zero foresight), t h i s proportion w i l l be set to that which maximises the return on new buildings. If they are expected to r i s e , then th i s proportion w i l l be set at that which maximises p r o f i t over the l i f e of the building. The length of l i f e of buildings i s therefore longer under conditions of perfect foresight than of zero foresight. Note that population growth here supplies the role of technical progress in economic growth models. Urban development in t h i s view i s therefore Harrod neutral. In the Evans approach the building stock is regarded as i n d e f i n i t e l y durable. The only cause for replacement i s r i s i n g land rents owing to population growth. Income and price e l a s t i c i t y of demand effects are ignored. In the Muth model, the impeti to redevelopment are r i s i n g incomes and deterioration of the building stock so that i t becomes progressively less suitable for the income group that rents i t . The decline in value however i s more closely related to r i s e in income than 97 physical deterioration, dwellings becoming too small r e l a t i v e to market demand to y i e l d a quasi—rent in excess of the opportunity cost of the land on which they are b u i l t . The developer seeks to maximise p r o f i t by setting the c a p i t a l intensity and length of l i f e of the dwelling at t h e i r optimum l e v e l s . Consumer preferences determine the type of housing that w i l l be b u i l t ; the developer has to reconcile t h i s with the rates of change of the various factors in the solution of the i n i t i a l supply—demand equation. An i m p l i c i t feature of Muth's model i s that residents agree with the developers assessment of the rate of change of the objective situation, so that he can maximise p r o f i t s over the length of l i f e of the development, rather than merely at the start of i t s l i f e . The question of d i f f e r i n g expectations as between producers and consumers is not considered again this is not a problem i f both parties have perfect foresight, so that t h i s model, too, smuggles in c r u c i a l assumptions of neo-classical economics by the back door. Rentals change over time as the r a t i o of the given building's housing services to the services produced by the consumer preferred building (given by the rate of growth of income and price of new stock, and the income and price e l a s t i c i t y of demand) changes. Ultimately, however, th i s w i l l decline to zero, at which point redevelopment occurs. Although t h i s i s a putty—putty model in which ex post improvements may be made, the developer selects those values of c a p i t a l intensity and length of l i f e of development so as to maximise value over 98 the entire l i f e of the building; thus no advantage is gained by redeveloping before rentals f a l l to zero. Muth then derives some predictions as to density of development using the same methods as Mi l l s ' work referred to e a r l i e r . However, where M i l l s obtains r e s i d e n t i a l density d i r e c t l y from ^ , so that (^lOJL(lO) Er\Uo' ^' a simple function of rentals only, Muth obtains for households, Ht , j i M - 4 L . J i L . „ e f r - O t = & h (^o-oweltynwnWoj where dL - e~£T) t £ = ^ + V/here / Qe^ eneoab) i-ate of grcwfc I arid supplied foiKe marM-<p - rate a^ufe l&rA rental* JW" - ( excxy\e<«ji^  fate o^jrovotfc. c^useKelds X - rate t^>wlk I V N I M Y ^ service so . H t . ^ e ^ V It i s far from clear then that new dwellings w i l l always be b u i l t to greater than average dens i t i e s . (Muth, 1975) This finding is in contrast with that of Evans who predicts that new development w i l l always be at a greater density. Muth also investigates what the population density in new 99 developments w i l l be as compared to what they would be i f housing were non—durable, and was thus new in any period. In the durable model, a variable occurs which i s not to be found in the non—durable model, and that i s the r a t i o of the discounts of the future values of c a p i t a l and land costs , and (ir-respectively, multiplied by the ratio of the discounts of future population and land in the area, ^ and respectively w i l l t y p i c a l l y exceed &r and so on this account, the durable model predicts lower land rent and higher aggregate population densities than the non—durable model. However, i f exceeds DY enough... the result may go the other way. Muth tests his model and finds that whereas non—durable models tend to over—predict average population densities, the durable c a p i t a l model he uses predicts population densities 20% higher than the estimates deriving from non—durable models. Evans provides no empirical testing of his model. These results appear a l l the more surprising in the l i g h t of the usual non—durable model prediction that growth due to increasing incomes takes place at lower and lower densities at the edge of the c i t y . However the reason i s not hard to f i n d . In part i t i s a result of the vintage theorists' roundabout discovery of the Ricardian margin. In geography, i t is more fami l i a r in i t s formulation by von Thunen, where i t is a result of transportation costs. At the extensive margin in the Ricardian or von Thunenlich concept, (agr i c u l t u r a l ) production i s land intensive. The further the distances to be t r a v e l l e d , 1 00 (say to and from market), the more land intensive production becomes. Travel distance i s a function of demand (as demand ri s e s more land w i l l be brought into production). If demand i s r i s i n g and entrepreneurs anticipate t h i s , and set productive techniques in anticipation of a r i s e in demand, then the length of t r a v e l associated with any productive a c t i v i t y at the start of any period w i l l be longer than i f they have zero foresight. Land therefore plays the role of c a p i t a l in the Ricardian/von Thunenlich formulation of the concepts of intensive and extensive margin, the other factor being land, or labour—with—capital, (this being the factor also that 'unlocks' the productivity of the land). In the c a p i t a l vintage model, the variables switch from being land, labour—with—capital and distance to being c a p i t a l , labour and time respectively. In the building stock version, these variables become c a p i t a l , land and time respectively. It is clear that by the time we reach t h i s point, that the position of land in the scheme has become completely switched around, and thi s i s why densities increase: under the assumptions of these models, c i t i e s cannot accomodate growth through expansion. In fact there has been a double switch of positions. In Ricardo, factor 2 , labour—with—capital, releases the productivity of factor 1, land; with c a p i t a l vintages, factor 1, c a p i t a l , enhances the productivity of factor 2 , land; with building stock vintages, factor 2 , land, releases the productivity of c a p i t a l . Furthermore, despite the fact that land reappears as a factor in the building stock vintage model, 101 space, and so also distance a f f e c t s , disappear from the picture. Therefore we can only deal with a single area with a uniform rental across i t . One of the appealing features of the vintage model i s that the c a p i t a l stock can be viewed as consisting "at any time of layers of f o s s i l s , or vintages, each set representing the amount of gross investment and the choice of technique under the p u l l of of expected factor prices and demand conditions of the time when the investment was made" (Harcourt, 1969) and these vintages stretching away in time, with the ones nearest us having the highest c a p i t a l i n t e n s i t y . This picture cannot be restated in sp a t i a l terms, however as building stock vintage theory reqires. Machines can exist and overlap in time. They cannot coexist or overlap in the same space at the same time. If the rentals in an area are the same (though indeed, t h i s i s the only way one area can be distinguished from another, since their building stocks w i l l d i f f e r under the assumptions of the model), then the buildings w i l l a l l be the same, have the same length of l i f e , and be demolished and replaced at the same time. There w i l l be no layers of buildings (except perhaps in the foundations), neither can new floo r s be added one at a time, as demand increases, as this would v i o l a t e the ex post fixed proportions conditions on which t h i s model is based. However the at—a—swoop demolition and replacement of buildings that the Evans and Muth models imply for an area is not far removed from the non—durable instantaneous adjustment models of urban growth, except that adjustment now takes place at discrete i n t e r v a l s , 1 02 not continuously. (This brings to mind Nuti's c r i t i c i s m of c a p i t a l vintage models, that since no labour i s required to transform putty into clay, that vintage models do not escape the assumptions and li m i t a t i o n s of the homogeneous c a p i t a l non—vintage models they attempt to replace — see also chapter 5.6). Evans attempts to show that i f the length of l i f e of vintages i s 9 years, a constant, and the area under consideration i s f\, the amount of land, L, devoted to any vintage i s therefore r\/8 , acres/year. Aside from i l l u s t r a t i n g the dimensionality problems involved in switching from the c a p i t a l to the building stock vintage models, ( i t is based on the argument that t o t a l area, f\ =\ L(v)dv i . e . , the sum Je-e ' of a l l the vintages in existence. In that case either area i s also a function of time, or . L- sT-^dV . As area i s fixed, then there can be only one vintage in existence), i t points up a clue that c a p i t a l or building stock vintage models are indeed dependent on homogeneous c a p i t a l concepts. It can be shown that an implication of the homogeneous c a p i t a l concept i s that the time pattern of labour inputs be constant (since for any homogeneous product, the capital/land ratios must be the same at a l l stages of production and the amount of c a p i t a l i s measured by the "roundaboutness" in production, i . e . , the amount of labour time locked up in the c a p i t a l good ). Here, i f area were allowed to be a function of time, the time pattern of land inputs to the urban growth process would be constant equal to In conclusion, when we r e c a l l that rationale underlying 1 03 urban economics is the existence of economies of scale and transport costs, a model of the r e s i d e n t i a l pattern of the c i t y which begins by assuming these away ( r e c a l l also that a l l production i s Cobb—Douglas) cannot be expected to t e l l us very much about t h i s process 1 (in fact, transport cost is considered to be i n f i n i t e l y high, as the c i t y cannot accomodate growth by expansion). The vintage model could provide an adjunct to a theory of r e s i d e n t i a l d i f f e r e n t i a t i o n , i f i t were coupled with an analysis of what causes land rentals in d i f f e r e n t parts of the urban area to r i s e at d i f f e r e n t rates from d i f f e r e n t i n i t i a l values, but i f such an account could be found, i t would render th i s approach largely superfluous. 'Muth's work appears to develop out of a finding reported in Muth(l968) that distance from the c i t y centre i s a variable which appears to l i n k age of dwelling and average reported money income. If the c i t y grows by accretion rather than by redevelopment, then location of housing i s an independent variable on the basis of which households are allocated. This l i n e of inquiry has been pursued in Muth (1977) . In t h i s case, growth of income, and deterioration in any area would explain the pattern of redevelopment there. 1 04 5 CRITIQUES OF NEO-CLASSICAL ANALYSIS: THE NEO-RICARDIAN ALTERNATIVE 5. 1 Introduction It should by now be evident that the issue of the use of homogeneous c a p i t a l concepts in neo-classical theory i s a controversial one. Just why i t is may not be so clear however, indeed i t is not clear to a l o t of neo-classical economists why i t i s either. To the c r i t i c s of the neo-classical approach however, i t symbolises the very essence of the shortcomings of neo-classical theory. To understand t h i s , i t i s necessary to be aware of the differences in approach to the role of theorising between the neo-classical school and i t s c r i t i c s . For instance Silberberg writes Let us consider three alternative hypotheses about the behaviour of firms. S p e c i f i c a l l y , suppose we were to postulate that: 1 Firms maximise p r o f i t s TI . . . 2 Firms maximise some u t i l i t y function of production U(T C )... p r o f i t s are desired not for their own sake, but rather for the u t i l i t y they provide the firm owner 3 Firms maximise t o t a l sales, i . e . , t o t a l revenue 1 05 only. By what means s h a l l these theories be tested and compared? It is not possible to test theories by introspection. The only way to test such postulates i s to derive from them p o t e n t i a l l y refutable hypotheses, and ultimately to see i f firms conform to the predictions of theory. (Silberberg, 1978, p. 12). Later he discusses u t i l i t y theory in these terms: U t i l i t y theory has been attacked on various introspective grounds... f i n a l l y i t might be argued by some that since u t i l i t y i s largely unmeasurable, any analysis based on maximising some unmeasurable quantity i s doomed to f a i l u r e . A l l the above c r i t i c i s m s are largely i r r e l e v a n t . The purpose of formulating these models i s to derive refutable hypotheses. In t h i s context, behaviour indicated by [ u t i l i t y maximisation] is asserted to be true for a l l consumers, [ u t i l i t y maximisation] i s our basic behavioural postulate. Refutation of [ u t i l i t y maximisation] can come about only i f the theorems derived from i t are demonstrably shown to be fals e on the basis of empirical evidence. (Silberberg, op. C i t . P. 25) The refutable propositions that neo-classical economics r e l i e s on, are of course, observations of marginal r e l a t i o n s : the reactions of decision variables to changes in parameter values. As the neo—classicals see i t then, the assertions that they make about behaviour and the assumptions they involve in order that refutable hypotheses may be produced for testing, are quite innocent. If they are fals e , then the hypotheses w i l l be rejected. However there should be no disagreement about what is being tested for: Positive economics is concerned with questions of fact, which are in p r i n c i p l e either true or f a l s e . . . Two economists, one favouring, say, more transfers of income to the poor, and the other favouring less, should s t i l l come to the same conclusions regarding 106 the e f f e c t s of such transfers. (Silberberg, 1978, p. 2) For the c r i t i c s of neo-classical economics, t h i s i s a l l a l i t t l e too ingenuous. To begin with, data rarely gives a true or false answer to refutable hypotheses, degrees of confidence are always involved. As a re s u l t , i f a hypothesis i s rejected, t h i s does not imply rejection of the theory, rather the assumptions are re—examined, and the hypothesis refined. Secondly, the question as to which facts constitute the relevant ones remains an open one, and w i l l in large part depend on the theory selected. For example, the economist favouring more transfers of income to the poor may recommend more equal ownership of the means of production: the economist favouring less may wish to look at transfer payments only. Even the same facts may be subject to a d i f f e r e n t interpretation depending on the theory employed. F i n a l l y , the type of theory employed determines the kind of questions that are asked. I f , for example, c a p i t a l is conceived of as a homogeneous quantity for the purposes of constructing a theory either of growth or d i s t r i b u t i o n , the questions that w i l l be asked are ones that concern the q u a l i t i e s of such a concept, which may or may not be of relevance to the operation of the real economy. Joan Robinson has c a l l e d t h i s homogeneous c a p i t a l concept leets, she writes of i t as follows: Many economists nowadays, who are in p r a c t i c a l questions are impatient of d o c t r i n a l disputes. 'what does i t matter?' they are i n c l i n e d to say. ' l e t him have his le e t s ; what harm does do?' but the harm that the neo—neo—classicals have done i s p r e c i s e l y , to 107 block of economic theory from any discussion of p r a c t i c a l questions. When equipment is made of leets, there is no d i s t i n c t i o n between long and short period problems... There is no such thing as the l e v e l of u t i l i s a t i o n of given equipment r i s i n g or f a l l i n g with the level of e f f e c t i v e demand... There is no room for imperfect competition competition.... There is no problem of unemployment. (Robinson, 1 970) Nuti (1970b) also argues against the neo-classical view view of theorising. Commenting on the position that u n t i l econometricians can provide the answer to the question of whether there i s enough s u b s t i t u t a b i l i t y in the system to permit testing the usefulness of neo-classical techniques, judgement of the neo-classical approach w i l l have to be suspended, he writes: The v a l i d i t y of any kind of econometric work, based on the notion of an aggregate production function i s undermined by [Sraffa's] c r i t i q u e . . . so that econometricians have no way of s e t t l i n g the dispute. Thus not only does the type of theory used determine the type of questions asked, but also the content of the answers. An analogy from the history of urban analysis i s the concentric zone hypothesis of Burgess. A strong argument can be made that t h i s model does not represent the theories of the human ecology school to any s i g n i f i c a n t degree, and that these theories do not depend on the v a l i d i t y of the model. Nevertheless, the model was taken as being an account of the theory. It thus required 'testing'. This sparked off 30—40 years 108 of debate, at least, as urban analysts sought to discover the precise nature of urbanism and urbanisation, the motive force of the model, and went looking for the answers empirically, in the s o c i o l o g i c a l investigation of urban l i v i n g . They also sought to give 'meaning' to the concentric zones. This meant finding something to put in them. F a c t o r i a l ecology studies provided family status. The real issue, that urbanism plays the same role in Park's vi s i o n of the c i t y as the Idea does in Hegel's vision of history, was not addressed, and the idealism inherent in the model meant that the answers from the data were bound to be inconclusive. Thus, selection of a theory is not an 'innocent' undertaking (Scott, 1975), especially in economics (despite the neo-classical belief to the contrary), where the phenomena that the theory seeks to explain include the d i s t r i b u t i o n of income between labour and c a p i t a l , growth and accumulation, which are also matters of v i t a l p o l i t i c a l and h i s t o r i c a l importance. In t h i s chapter we review the c r i t i q u e of neo-classical economics. We begin by considering the c l a s s i c a l school, out of a c r i t i q u e of whose doctrines neo-classical economics evolved, and then the post-Keynesian and neo—Ricardian c r i t i q u e of the neo-classical school. 109 5.2 C l a s s i c a l Economics The c l a s s i c a l economists, of whom Marx may be considered the leading figure, regarded economics, that i s , analysis of price-output r e l a t i o n s , as but a part of p o l i t i c a l economy. They conceived of an economic system where the t o t a l surplus, that i s , the ra t i o of gross output to means of production was a technologically determined maximum, but where the shares of wages, rents, and p r o f i t s in t o t a l surplus were not. The analysis of these r e l a t i v e shares in t o t a l surplus was the focus of their p o l i t i c a l economy. Capital and land were regarded as aids to production, but the ultimate cause of productivity and source of value was labour. This i s the basis of the theory of economic rent, where i t is stated that high value cash crops are not grown on a piece of land because rentals there are high, but that land rents are high there because cash crops are grown on them. The cause of these cash crops being of high value i s the amount of labour, or, in Marx's refinement of the . concept, of labour power involved in their production. The p o l i t i c a l implications of such an analysis are cl e a r . If i t i s labour that creates value, including the value of the land and of c a p i t a l , what right do landlords and c a p i t a l i s t s have to any part of i t ? The rebuttal of th i s argument concentrated on the d i f f i c u l t i e s of putting a figure on labour value, es p e c i a l l y should wage and p r o f i t rates a l t e r . While i t was conceded that in an a g r i c u l t u r a l economy where the surplus produced annually was consumed annually, prices might well be a r e f l e c t i o n of 1 10 labour values, in any other type of economy, for example, one in which a surplus is carried over into the following year, or one in which durable machinery was employed in the various sectors, a change in the relation of wages and p r o f i t s would aff e c t d i f f e r e n t industries d i f f e r e n t l y . If we said that the value of output was that given by the cost of labour at the time that sector was brought into production, and now the wage share (=value) of labour in f i n a l output has been reduced, then prices have diverged from labour values and unless a way of reconciling the two movements can be found, the assertion that labour values determine prices cannot be proven and so may be rejected (Howard and King, 1975). It was Marx's f a i l u r e to deal with t h i s "transformation problem" that was the great stumbling block to acceptance of his analysis of capitalism. For Marx, c a p i t a l had two aspects: while i t was an aid to production, i t also dominated production, and i t was the issue of how and why that dominance exerted i t s e l f that concerned him. He wished to show that c a p i t a l i s t production was not only unfair, but i n e f f i c i e n t and unnecessary. If he could not show that labour power was indeed the determinant of value, and that the price system reflected t h i s , then a v i t a l part of his argument was missing. He (or rather Engels) posthumously published a solution in Vol. I l l of 'Capital! , but i t was quickly discovered that his solution depended on the assumption that capital/labour ratios in a l l sectors were i d e n t i c a l . In such a case, changes in the rate of p r o f i t would 111 a f f e c t prices at a l l , and in fact his solution seemed to assume away the problem. (Harcourt, 1972) 5.3 Neo-Classical Economics And Marginal Theories Of D i s t r i b u t i o n : Discussion And Critique Partly because of the i n t r a c t a b i l i t y of the labour theory of value problem, and partly because of their wish to avoid the p o l i t i c a l questions i t s adoption entailed, neo-classical economists decided to approach the question of value from the analysis of value in exchange, or u t i l i t y . Here i t was marginal r e l a t i o n s , or changes in quantities demanded with changes in price that became the focus of attention. This work developed as an analysis of demand and exchange before i t was applied to an analysis of production, along the lines proposed by Wicksteed which reviewed e a r l i e r . It i s worthwhile noting before proceeding further that demand relations do not play any s i g n i f i c a n t part in the c l a s s i c a l picture (cf. Robinson, 1961). Prices of the wage goods and of the commodities needed to produce them are determined exclusively by the relations between wages and p r o f i t s , thus placing exchange relations firmly within the sphere of productive r e l a t i o n s . Wages are given as a s o c i o — h i s t o r i c a l datum. Luxuries are any other commodities not consumed by a l l 1 1 2 labourers. Though their t o t a l quantities are fixed by the rate of p r o f i t , their r e l a t i v e prices are indeterminate with respect to, and do not affect the rate of p r o f i t . Only once their r e l a t i v e t o t a l s are given, w i l l supply—demand relations a f f e c t their r e l a t i v e prices (Pasinetti, 1974). Thus demand relations are important for "the d i s t r i b u t i o n of income not between the haves and havenots, but between the haves"(Nuti, 1970a), since demand relations only affect the prices of luxury items. This situation i s also the o r i g i n of Kalecki's famous, i f apocryphal, remark, that Workers spend what they get; c a p i t a l i s t s get what they spend. It i s also the the o r i g i n of Keynes "widow's cruse" theory of d i s t r i b u t i o n (no matter how much was poured out of i t , i t was always found to be f u l l ) , and thus indicates the the closer l i n k s Keynes has with the c l a s s i c a l s than the neo—classicals. Dobb(l940) commented on t h i s phenomenon: Von Bortkeivicz rubbed in very aptly the c o r o l l a r y of Ricardian determination of p r o f i t by the conditions of production of wage goods:namely that 'the o r i g i n of p r o f i t l i e s in surplus labour... i f i t i s indeed true', he wrote 'that the l e v e l of the rate of p r o f i t in no way depends upon the production of those goods which do not enter in to real wages, then the o r i g i n of p r o f i t must c l e a r l y be sought in the wage relationship and not in the a b i l i t y of c a p i t a l to increase production. For i f this a b i l i t y were relevant here, then i t would be inconceivable why c e r t a i n spheres of production should become irrelevant for the 1 1 3 question of the " l e v e l of p r o f i t ' " . Neo-classical economics, by contrast, makes production relations depend upon the sphere of exchange relations, including production r e l a t i o n s . This leads the neo-classical school into an inconsistency. Robinson(1961) writes: In a market economy there may be a tendency towards uniformity of wages and the rate of p r o f i t in dif f e r e n t l i n e s of production, or prices may be governed by supply and demand, but not both. Where supply and demand rule, there i s no room for uniform levels of wages and the rate of p r o f i t . . . The intrusion of demand equations into the theory of the wage economy, and the attempt ot f o i s t a rate of p r o f i t onto the price economy have led to endless confusion: a c r i t i q u e to clear i t up i s long overdue. Returning to Wicksteed, r e c a l l that in an e a r l i e r section we argued that Wicksteed produced a theory of rewards to c a p i t a l , faute de mieux, through the application of Euler's theorem. This view i s confirmed by Dobb(l940): If S=x+y+z, and x+y are given, i t must necessarily follow that z=S-(x+y). Such a mathemathical truism, said Wicksteed, could equally be applied to x and to y as to z. On the same li n e of reasoning [that land rent i s a 'residual surplus'] the price of c a p i t a l or the price of labour could be treated as a residual surplus. It was a l l a matter of which factor was taken as 'given', and which as the residual variable, was determined. This demonstration of the existence of a marginal product of c a p i t a l c l e a r l y involves a l o g i c a l sleight of hand. Three issues present themselves: the v a l i d i t y of the a t t r i b u t i o n of d i f f e r e n t i a b l e q u a l i t i e s , present in land, and labour—with—capital, to c a p i t a l alone; the v a l i d i t y of assuming 1 1 4 that x, y and z are as interchangeable in r e a l i t y as they are in the algebra, in particular the treatment of labour on a common footing with other factors of production and the character of the c a p i t a l component in the production function. We consider each in turn. In the c l a s s i c a l system, one can talk about marginal increments of land and observe the reaction of output to such an increment, or even marginal increments of labour—with—capital e.g., the application of f e r t i l i z e r , but the notion of a 'marginal productivity of c a p i t a l ' i s an unnecessary one in c l a s s i c a l economics. This is because i t i s the surplus that i s determined by technical conditions of production, not the rate of p r o f i t . To state with neo-classical economists that i s to assert that there also exists a production function (homogeneous of degree one). Note however that the size of the surplus cannot be predicted from the form of the production function. In neo-classical economics, the size of the surplus becomes an open question, but the d i s t r i b u t i o n of income i s now techn i c a l l y determined by the c h a r a c t e r i s t i c s of the production function. The p o l i t i c a l implications are subtle but cle a r . Wages move from being an index of value to being one of cost. Value i s determined not by production, but by sale. Labour value derives from the value—in—sale of the product, not from the value of labour power in the productive process. Value i s maximised when 1 15 p r o f i t i s maximised, not when the t o t a l surplus i s maximised. Attempts to change the d i s t r i b u t i o n of income via trade union pressure for instance are self—defeating, since t h i s would only breach the technical conditions of production, and cause i n e f f i c i e n c y and reduced output. A grab for a larger s l i c e of the pie only reduces the size of the pie, so nothing is gained. S i m i l a r i l y , the object of economic analysis i s no longer p o l i t i c a l economy, the analysis of the conditions of production and of the d i v i s i o n of the surplus, but economics, the analysis of the conditions when, with 'given' factor supplies and 'given' prices, p r o f i t s and/or u t i l i t y w i l l be maximised. In fact the two situations are complementary, u t i l i t y maximisation e n t a i l s p r o f i t maximisation. The market solution therefore supplies the greatest happiness to the greatest number. Production constraints are conspicuous by their absence. Neo-classical economists attempted to r e c t i f y the i n a b i l i t y of the production function to say anything about the l e v e l of output by recourse to Say's Law, i . e . , that supply creates, or always ri s e s to meet i t s own demand. If there were productive factors in existence, demand would always keep them f u l l y employed so long as they supplied their services at the market clearing p r i c e . If they persisted in charging above equilibrium prices, they would become unemployed. For Say's Law to operate, a l l markets must clear at the equilibrium pr i c e . However, the wage of labour does not clear the labour market but in fact clears the goods markets (Solow and S t i g l i t z , 1968). The real 1 1 6 wage of labour i s thus not the money bargain struck in terms of an equilibrium between the labour supply function (the marginal d i s u t i l i t y of work) and the labour demand function (the technically determined marginal product of labor), but the money wage divided by the price l e v e l (Keynes, 1936). The labour market is cleared when aggregate consumption and investment reach the l e v e l of f u l l employment output. If consumption i s a function of income (and thus dependent on the l e v e l of output), then the l e v e l of employment depends not on the money bargain struck by labourers, but on decisions, made by entrepreneurs (ones which are, in a c a p i t a l i s t system, inevitably subject to uncertainty), as to the le v e l of investment (which requires c r e d i t to finance i t u n t i l revenues are real i s e d in sales). This le v e l of investment and thus also employment i s therefore set by the relations of investors' rates of time discount to money rates of interest (this i s the gi s t of Keynes' arguement in his "General Theory of Employment, Interest and Money"). The fact that the wage of labour does not clear the labour market makes labour a unique input into the productive process. Nuti(l970b) writes: Of a l l commodities, labour power i s the one for which in a c a p i t a l i s t system there no forward markets: workers—unlike bondsmen, slaves, horses and robots—can leave their jobs whenever they l i k e . No forward committment in terms of labour time can be enforced; t h i s not a market imperfection; on the contrary i t i s a necessary condition for a perfect labour market in each period. The symmetry between production and exchange might be i n t e l l e c t u a l l y pleasing, but the price to be paid for thi s generalisation i s tremendous, namely the i n a b i l i t y to 1 1 7 understand the presentday economy where labour power is a commodity and production is a r e l a t i o n among men as well as with nature. On a more mundane l e v e l , Joan Robinson has been concerned to deny the v a l i d i t y of the doctrine that wages are regulated in the short run according to marginal productivity theory. The wage of labour, she argues, w i l l vary anywhere from zero to average output/head depending on the l e v e l of u t i l i s a t i o n of equipment. In long run equilibrium, wages, technical conditions and p r o f i t s w i l l be in adjustment with each other, but i f technical conditions have adjusted to the rate of p r o f i t , the p r o f i t rate's l e v e l i t s e l f becomes indeterminate. F i n a l l y , we come to the nature of the c a p i t a l component in the production function. If d i s t r i b u t i o n i s to be determined by the marginal products of the production function, i t i s v i t a l that there be no ambiguity in the results, otherwise some other cause must be responsible and some other explanation the correct one. The production function which also describes how c a p i t a l i s exchanged for labour must be 'well behaved' i . e . , smoothly d i f f e r e n t i a b l e with respect to a l l of i t s arguments, including especially the c a p i t a l component. This is why c a p i t a l must be regarded as homogeneous, malleable and instantaneously and c o s t l e s s l y transformable into any commodity, either c a p i t a l good or consumption good at any time. The c a p i t a l good must be homogeneous with wage good in order that they can be both measured in the same units. It must be instantaneously and c o s t l e s s l y transformable so that the c a p i t a l to labour r a t i o can 1 18 respond instantaneously and c o s t l e s s l y to changes in wages and p r o f i t s , so that constant shares in net output may be maintained, whatever the r e l a t i v e l e v e l s of wages and p r o f i t s . This means that as the p r o f i t rate f a l l s , the r a t i o of c a p i t a l to labour in production must rise in perfect reaction to the changes in p r o f i t rate. As only one technique i s associated with any given c a p i t a l to labour ratio a f a l l in the p r o f i t rate also leads to higher output/head. This homogeneous c a p i t a l model i s a good example—in fact i s the perfect example—of the differences to—ward theorising held by the neo—classicals and their opponents. By our comparison of the c l a s s i c a l and neo-classical schools of thought we have demonstrated the untoward p o l i t i c a l consequences of adopting the neo-classical approach. For the neo-classical economist however, this model is a mere device for producing refutable hypotheses. The defence of the neo-classical use of homogeneous production functions such as the one described by Samuelson(1962) and i t s c r i t i q u e by Garegnani(1970) i l l u s t r a t e p e rfectly this approach. We introduce this debate with a discussion of Sraffa's ( i m p l i c i t ) c r i t i q u e of neo-classical theory. 1 19 5.4 The Sraffa System We discussed e a r l i e r how c l a s s i c a l economics foundered on the rocks of the c r i t i c i s m that an invariant standard of value could not be found. This standard was discovered by Sraffa and presented in his "Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory" (1960). Like the c l a s s i c a l economists, Sraffa makes the surplus technologically determined, and leaves the wage—profit re l a t i o n open. Production in this system consists of a set of linear price equations foMaV •••- tRAK'+r) +L„y =P„A • • • where A^D^fcd. i s the proportion of commodity I that i s used in the production of f\,£ etc., |_ft, L_B i s the labour input to ft , are the prices of commodities (\ ,B and w and r are wages and p r o f i t s , respectively. This set of H equations contains K.+1 unknowns and so moves with one degree of freedom. If therefore, the value of the output of one sector, usually the wage good sector, i s set to 1 the system w i l l be determinate. The problem is to f i n d a way of expressing the relation between T and W which w i l l not vary as they themselves vary. The solution i s to find, or construct, a system of equations from the data presented here such that 120 The various commodities are represented among i t s aggregate means of production in the same [ratio] as they appear among i t s products. Furthermore, th i s ratio must also occur in a l l the succesive layers of the industry's means of production without l i m i t . (Sraffa, 1960). Such an industry or construct is the Standard System. It i s invariant to changes in the p r o f i t s and wages, because a l l the price changes that do occur balance out exactly. The net output or surplus equals gross output minus means of production and the ra t i o net output/means of production represents the technologically determined rate of surplus, , in the economy, and is unaffected by changes in wages and p r o f i t rates whose rela t i o n i s expressed in terms of i t , since wages are a share in net output (Howard and King, 1975): r = R ( i - w ) . To construct the Standard Commodity, Sraffa needs a set of mu l t i p l i e r s n V f t f ^Q'^c t o t o a p p l v t o t n e P r i c e equations in such a way that a = 3 - K = 1 + R etc The units in which the m u l t i p l i e r s are to be expressed are defined by giving k-+l equations and K+l unknowns (the k m u l t i p l i e r s and 121 r\). Sraffa denotes the mu l t i p l i e r s thus found as CJ/ft, and applies them to the set of price equations to define the Standard system, etc. A d i s t i n c t i o n may be made between Basic commodities which enter into the production of every good and Non-Basics which do not enter into the production of at least one good. Only Basic commodities are required to construct the Standard System, the prices of the Non—Basics do not a f f e c t the rate of p r o f i t , or the wage rate. They are equivalent to the luxury goods mentioned previously, except that as opposed to the c l a s s i c a l and Marxist t r a d i t i o n , in which the concept of luxury goods was discussed, Sraffa treats wages also as a Non—Basic so as to avoid arguments about the nature of a subsistence wage. A similar approach which treats wages as a 'Basic' good is that of von Neumann, (1945). It i s important to bear in mind that t h i s system i s a system of price equations only. The standard of value is a set of modified price equations, and since, as Robinson (1961) puts i t , there are prices, but no markets, and wages, but no pay packets, values, derived either from production, or from exchange, nowhere make an appearance. The purpose of th i s system, rather, i s to demonstrate how certain previously intractable problems in c l a s s i c a l economics may be solved; namely those problems concerned with the construction of an invariant standard of value by means of which 122 the e f f e c t s of changes in wages and p r o f i t s on r e l a t i v e prices might be assessed. So far we have considered only the case of simple reproduction with c i r c u l a t i n g c a p i t a l goods, with no joint production or alternative processes available to produce the same good. The case of jo i n t production includes that of fixed c a p i t a l since each productive employing fixed c a p i t a l can be considered a case of joint production of output plus a used machine. Alternative processes w i l l be characterised by di f f e r e n t levels of output for any given value of W and r . In general, a l l that can be said about the re l a t i o n of W to P at constant output is that as T increases, W decreases, but for any given r , 4w/^r \ 0, depending on the time pattern of inputs into the process involved: Think of the cost and f i n a l price of a commodity as being arrived at as the summation of a v e r t i c a l series of stages of production spread out backwards in time, each consisting of a labour input plus commodity inputs (machines, raw materials) that are products of some e a r l i e r stage; each with i t s labour inputs having i t s attached date in the v e r t i c a l series. Manifestly, everything w i l l depend on how those labour terms are dis t r i b u t e d in time. Suppose two commodities, one with larger t o t a l labour inputs, but these bunched at recent dates, and the other with much smaller t o t a l labour inputs bunched at distant dates. With low wages and high interest, the f i r s t may come out cheaper despite i t s larger wage b i l l . As wages r i s e and interest rates f a l l , the second w i l l have the advantage because of i t s lower wage b i l l : an advantage that one would expect that one would expect i t to retain however high wages rose and interest rates f e l l . This is the orthodox case to which an ordinary production function can be f i t t e d . But suppose a case where one commodity has a l l or most of i t s labour inputs supplied at some intermediate date, the other one having some labour at a very distant date, but the 1 23 bulk of i t at a quite recent date. It i s perfectly possible for the second of the pair to have the price advantage at intermediate lev e l s of interest and wages, but the f i r s t one to be preferred (because cheaper) both at very high lev e l s of interest (with low wages) and at very low level s of interest (with equivalently high wages). The reason i s the p o s s i b i l i t y of differences in the compounding eff e c t of interest rate changes on the comparative cost of inputs of very distant and of intermediate dates... Another way of expressing i t would be to say that i t depends on widely d i f f e r e n t proportions of labour and other inputs at succesive layers of production. (Dobb, 1970a) Consider f i g . 5.1 below, graphing the wage—profit r e l a t i o n for two techniques o< and , producing the same commodity. The segments Ow^ and Ov'p measure the net physical product/worker, the maximum wage obtainable when r=0. The segments Ov^Owp measure the amount of net physical product/worker that goes to wages at interest rate , VotV/'oi and VI^V^ measure therefore the amount that goes on interest payments at the rates rt and r% respectively. The value of capital/worker in- <X - - *Q^* = = tan/,wjw£. Similar i l y for (3> i t i s tanLwpT v/'(J . Clearly i f the wage rate is set at w^ , then technique ^ gives a higher rate of return than does technique c* (Yi>rju ). However t h i s r e l a t i o n is symmetrical, so that i f the interest rate is set at r\ , then technique <* , y i e l d i n g a higher wage rate w i l l be chosen. The economic interpretetion of the curvature of <* and fi is that, i f each of these techniques can be regarded as being composed of two processes, the f i r s t producing the machinery needed to produce the output, and the second using the machinery to produce the output ('capital' good and 'consumption' good FIGURE 5.1: Pair Of Techniques E x h i b i t i n g D i f f e r i n g Wage-Interest Relations 1 25 sectors, respectively: note that the existence of least two sectors i s v i t a l , otherwise the output would be produced by i t s e l f and labour, i . e . , the c a p i t a l and consumer goods would be homogeneous.), then the curvature of these curves is a function of the capital/labour r a t i o s (' K / L =K ) in each sector ( ' K ' and 'C ', respectively) of each technique. In the case of technique Ci , so that a r i s e in the value of r affects the c a p i t a l good sector proportionately more than i t does the consumption sector. Therefore a r i s e in the rate of interest w i l l cause the value of capital/head to r i s e at an ever increasing rate producing (to the origin) wage—interest trade—off. Conversely Kc produces a convex curve, and of course i f k K=k c, the wage—interest relation w i l l be a straight l i n e . It i s also important to bear in mind that f i g . 5.1 i s a l o g i c a l construct only. It shows which technique would be chosen in the circumstances described by the wage—interest curves foroc and (3 i f the rate of interest were at such and such a l e v e l . It cannot be used to in the description of a process occurring over time. This why the terms interest and rate of return are used here rather than p r o f i t , as p r o f i t i s an estimate of future returns, the si t u a t i o n examined here i s comparing actual wage rates with actual returns. This graph demonstrates that technique oC i s associated with ranges of the wage rate as the one y i e l d i n g the highest rate of return. It cannot be used to say that as wage and interest rates change over time, f i r s t (X, 1 26 then {3 r then oC again w i l l be chosen. The time axis is perpendicular to the plane of the V , r axes, and the wage—interest r e l a t i o n may a l t e r along that axis in ways unknown at present. Neo-classical economists have been much less cautious about using such graphs to describe processes over t ime. With th i s warning in mind, l e t us examine some of the properties of f i g . 5.1. The most important feature i s indeed that technique oC is associated with two d i f f e r e n t ranges of the rates of interest, and/or the wage rate. Therefore the technical conditions of production do not y i e l d a d e f i n i t i v e statement about the d i s t r i b u t i o n of income, and the same technique i s associated with sets of values of capital/head. Furthermore, t h i s result does not depend on any assumption about the c a p i t a l good. In the situation described here, the c a p i t a l goods used in each technique are in fact homogeneous, produced by themselves and labour, though in the technique as a whole there are heterogeneous goods. If we were to. disaggregate the c a p i t a l good sector ( f t x . . . K n ) u n t i l a f u l l y nonhomogeneous system could be described, the W-tv trade—off would be much m more bumpy. The number of changes in sign of dw/dr would be given by the number of a l t e r a t i o n s of sign of K k ~" K u (Nuti, 1970a). The economic interpretation of such bumps i s that they indicate, over some range of the rate of interest, the firm using that technique i s involved in borrowing and lending operations, borrowing in some time periods and lending in others and that i t would gain more from an increase in the interest rate as a borrower than i t loses as a lender, so that i t i s able 1 27 to pay a higher wage rate, i f i t can perform lending and borrowing operations, at a higher interest rate This result derives from another c h a r a c t e r i s t i c of the Sraffan system, that when there i s no fixed c a p i t a l , disaggregation by sectors or over time periods i s exactly equivalent. Each set of price equations can in p r i n c i p l e be reduced to a set of dated quantities of labour, "labour inputs/unit of output with time period attached to these expenditures, seeing that the rate of p r o f i t i s involved in price determination" (Dobb, 1970a), providing that the l e v e l of real wages (Vl/p , p =price level) has been set. The technical reason that the case of joint production does not permit the reduction of output into quantities of dated labour i s associated with the fact that under conditions of joint production, non-basic goods are produced along with basic goods which must be "removed" from consideration in order to construction. In order to construct the standard system, th i s w i l l involve negative m u l t i p l i e r s . The Standard System i s thus no longer a "conceivable re—arrangement of the actual processes" (Sraffa, 1960) but an abstract construct. Also some prices may be come negative, processes associated with negative prices would then have to be discarded. Reduction of output into dated quantities of labour produces terms with negative values, to which i t i s impossible to give an economic meaning. In economic terms t h i s has the effect that The inputs of di f f e r e n t dates j o i n t l y produce the outputs of diff e r e n t dates; and i t i s impossible to 1 28 separate out the contribution to the output of d i f f e r e n t dates of the input of the input of a single date. (Kaldor, 1937) So therefore, in the case of f i g . 5.1, we have assumed that as well as as having a homogeneous c a p i t a l stock, there i s no joint production, therefore no fixed c a p i t a l to complicate the analysis. These two assumptions the neo-classical production function also makes. This permits more dire c t comparison of the two approaches. In the outline of his system, Sraffa makes nonproduced means of production non—basics, thus raw materials, land and also wage goods, are non—basics in Sraffa's scheme and do not a f f e c t r e l a t i v e prices or the wage-profit r e l a t i o n . As mentioned above, Sraffa's reason for making wage—goods non—basic i s to avoid implying that the wage was a te c h n i c a l l y determined subsistence one, which would have followed from designating them as basic. The wage goods could have been s p l i t into a portion which were basic and a portion which were not, but he f e l t that t h i s would c o n f l i c t with our i n t u i t i v e concept of the wage. Thus the wage bargain has to be set by s o c i a l action. As mentioned above also, t h i s i s his major break with c l a s s i c a l analysis where the concept of a te c h n i c a l l y determined subsistence wage, 'The Iron Law of Wages', did much to create the label of economics as the 'dismal science'. Along with the question of the necessity of a labour theory of value, t h i s issue i s the p r i n c i p a l source of contention between the neo—Ricardians and the Marxists. 129 However Sraffa's system's greatest virtue i s in i t s f l e x i b i l i t y and there i s no reason why i t cannot be used on behalf of Marxist analysis, as indeed i t has been (e.g., Meek, 1967, 1973). Sraffa, however, makes absolutely no comment on these issues. As he .presents i t , i t i s merely a way of expressing in a definitve consistent and p r a c t i c a l manner, solutions to the problems of p r i c i n g and valuation of c a p i t a l goods and outputs when wage—profit relations a l t e r . Incidentally, though t h i s is in fact the u l t e r i o r motive, i t provides the basis for a severe c r i t i q u e of neo-classical theory. The wage—profit r e l a t i o n , as graphed for example in f i g . 5.1, provides the ground on which the two theories may be compared. There are three issues on which the Sraffa analysis d i f f e r s from, and may be said to be superior to the neo-classical analysis (Robinson, 1961), since i t has to make far fewer assumptions about the nature of the productive process, the only one of any import for an analysis of real world conditions, that of constant returns to scale, i s one shared with neo-classical economics. The f i r s t point i s that i f constant returns to scale apply, demand equations are irrelevant to the establishment of prices in a society characterised by competitive production (which tends to equalise the p r o f i t rate between sectors—abolition of demand as a parameter does not necessarily a l t e r any assumptions about competition. However the analysis would be unaltered i f interest rates were set by planners). This returns one quite d e f i n i t e l y to the c l a s s i c a l 1 30 world, where relations in production determined d i s t r i b u t i o n and prices. The second point is that the basic assumption of Wicksteed's formulation of marginal productivity theory, derived from Adam Smith, namely that "the price of any commodity, either immediately or ultimately resolves i t s e l f e n t i r e l y , (that i s to say, without leaving any residue) into wage p r o f i t and rent" (Sraffa, op.cit.) must be rejected: i f there is joi n t production, this i s simply impossible; i f there i s simple reproduction only, price resolves i t s e l f into a series of dated labour inputs. F i n a l l y the marginal productivity theory of d i s t r i b u t i o n i s based on a false premiss, the asumption that production functions exist and not only describe marginal rates of substitution between factors, but also account for income d i s t r i b u t i o n . That this premiss i s false can be seen from f i g . 5.1 alone, where one i s associated with two d i s t i n c t ranges of the wage—interest r e l a t i o n , and thus cannot be associated with a single d i s t r i b u t i o n of income. Sraffa's system, to—gether with the (post—)Keynesian c r i t i q u e of neo-classical economics' labour market theory, and the theory of e f f e c t i v e demand (in which the le v e l of investment, not technical conditions, determine p r o f i t s ) provide not only a powerful c r i t i q u e of neo-classical economics, but also a basis for an alternative paradigm (e.g., P a s i n e t t i , 1974). 131 5.5 Samuelson vs. Garegnani Neo-classical economists, of among whom Samuelson and Solow may be regarded as the leading figures, were not slow to respond to this assault on the foundations of their theories samuelson's response was to admit the logic and v a l i d i t y of the arguments against neo-classical theory, but to deny their importance (Samuelson, 1962). He set up a series of two-sector Sraffa equations, in which sector 1 produces, by means of i t s e l f and labour, the c a p i t a l good for use in sector 1 . Making the consumption good a numeraire i . e . , p =1, he price equations of t h i s integrated capital—consumption good industry: writes where 0C-L^ i s the input/output c o e f f i c i e n t of commodity L into commodity ^ ( i , j , = 1, 2). Dividing (5.5.1) by X A and (5.5.2) by Xj, gives the simple or, r e l a b e l l i n g for convenience p> K(l+r) + wlK- = PK where. = X L , g~ .11 , K= n CI = 0,1,1) 5.5. W 1 32 and r\ and C refer to c a p i t a l and consumption good sectors respect i v e l y . The wage rate in (5.5.3'), (5.5.4') may be expressed as a function of the p r o f i t rate X - M M r ) £4*5.5.5. Equation (5.5.5) provides the formula for W in terms of T that is the basis of the graphs drawn in f i g . 5.1. Obviously, varying c a p i t a l to labour ratios in each of the two sectors w i l l provide a whole family of curves for a given output, ranging from very concave to very convex. Samuelson proposed to r e s t r i c t his analysis to "a r e a l i s t i c subclass of techniques", ones in which output/head remained constant, but which d i f f e r e d by virtue of their L r a t i o s . He then showed that the outer envelope of wage curves (christened the factor—price frontier) obtained from this heterogeneous ensemble of techniques, gave a l l the same properties of, and thus, as c l o s e l y as one wished, could be approximated to, by a neo-classical malleable c a p i t a l aggregate production function. Thus proving yet again that debate over the assumptions of neo-classical theory was irr e l e v a n t . The theory is c e r t a i n l y mythical, nonetheless i t serves i t s purpose of producing refutable hypotheses. Garegnani(1963, and more f u l l y 1970) was not about to l e t Samuelson get away with such an argument. His immediate l i n e of attack was to go after Samuelson's assumption of constant output/labour ratios in a l l techniques. Comparison of f i g . 5.2, 133 Samuelson's f i n a l solution with f i g . 5.1, provides the clue. As can be seen, the outer envelope does indeed approximate the shape of the isoquants associated with neo-classical aggregate production functions. However, from the discussion of the wage curves in f i g . wage—profit r e l a t i o n s . Straight wage-profit relations in turn imply the same c a p i t a l to land r a t i o s , in both sectors of production of each technique. This implies also that the time pattern of labour inputs i s constant throughout each technique. This i s in fact the case where Marx's transformation of labour values into prices applies d i r e c t l y , as the r e l a t i v e pattern prices of the commodities in each sector are therefore inpependent of the rate of p r o f i t . It also the case in which Bohm—Bawerk's theory of c a p i t a l (average period of production=interest rate) as a pure aid to c a p i t a l applies. Furthermore, when we r e c a l l that the c a p i t a l is homogeneous with i t s e l f , and the only technical information distinguishing the c a p i t a l good sector from the consumption good sector is the d i f f e r i n g proportions in which c a p i t a l i s combined with labour in the two sectors (to produce either the consumption good or the c a p i t a l good), i t i s clear that where the capital-labour ra t i o s are the same in both sectors, that the c a p i t a l good must be homogeneous with the wage good. Therefore Samuelson never leaves the confines of a malleable c a p i t a l world, and the fact that the neo-classical alternative to Marx's d e f i n i t i o n of 5.1, r e c a l l that constant ratios imply straight l i n e FIGURE 5.2:Factor P r i c e F r o n t i e r of Techniques w i t h Constant R a t i o s , Ordered by Ra t i o s . ( a f t e r Samuelson,1962) 1 35 c a p i t a l r e l i e s on the same assumptions as Marx's own analysis does have a certain irony to i t . In fact, capital—labour r a t i o s w i l l generally be d i f f e r e n t between d i f f e r e n t sectors of production, as in each technique in f i g . 5:1, and the s t a t i c analysis of price determination can then be handled only by a Sraffa system. The neo-classical production function w i l l not be able to give the technical s p e c i f i c a t i o n s , nor an account of the d i s t r i b u t i o n of income in terms of the interest rate. Reswitching (the reappearance of a technique on the factor—price frontier at a low rate of interest when i t also appears there at a higher, but separate, rate of interest: technique oC in f i g . 5.1) i s not necessary to prove t h i s either. Capital reversing, a switch from a higher to a lower c a p i t a l to labour r a t i o technique as the interest rate f a l l s w i l l also do the t r i c k . Such a reverse occurs at the interest rate r^p in f i g . 5.1. , however, is a conventional c a p i t a l movement. Suppose that the s i t u a t i o n graphed in f i g . 5.2. did apply. Could we then say that in t h i s case at least, the Marxist and neo-classical procedures were equivalent? As a f i n a l nail, in the c o f f i n of the neo-classical production function, Garegnani shows that not even this consolation i s l e f t to neo-classical analysis. The problem can be restated as follows. We are asked to define a function S =S(3,L) homogeneous of the f i r s t degree — with |§ the quantity of (net) product, C" that of c a p i t a l , and L, that of labour — s a t i s f y i n g the following conditions two conditions: 1)^S/QL = e^S/dO"), where W= e(r) i s the r e l a t i o n 1 36 between w and T in the 'real' economy. 2) S /L = ^(fcS/dtf), where (j,. =\{T) is the rel a t i o n between net product/worker and r in the 'real' economy. The function $( J , L ), i f i t existed, would be •'vr a surrogate production function, for the 'real' economy in the sense that i t would determine the re l a t i o n between w, r and once J" — the quantity of surrogate c a p i t a l — has been appropriately defined... To see whether the surrogate production function can be defined, we begin by using Euler's theorem to write S(7 , L ) in the form L 6L do" L ^ in the imaginary economy, the equilibrium rate of interest r would always be equal to 6SAJ'. Condition 1 then permits us to write function [ 5.5.6]in the form JL = e(r) + r(|.) 5.5V D i f f e r e n t i a t i o n of function [5.5.6] with respect to dSldJ gives d(S/Q d f r / Q _ dCoS + G ; dCSk) d(3/U d(6S|6J) dfoSfcjcT L 60" "d&SldO"). and then since d(S|L} - ^jj. w e obtain X - - dtoS /dL) * ~' cl(6S/6o-) and using condition 1 again [5.5.7 rules out any cases of a wage—interest r e l a t i o n that i s concave to the o r i g i n . S( <J* , L ) i s thus not a production function for those cases. However, convexity in the v—r r e l a t i o n i s not a s u f f i c i e n t condition for i t s being a production function either.] 1 37 To find [the s u f f i c i e n t condition], l e t us return to function 5 . 5 . 6 ' and using result 5 . 5 . 7 rewrite i t in the form | = -e(r) + r(-e'(r)] It i s now clear that condition 1 above i s s u f f i c i e n t to define the function S( J , " L ) . No freedom is l e f t for adapting S ( J , L ) so as to adapt to condition 2 . We can only ascertain whether whether S/L as defined by [ 5 . 5 . 7 ] i s i d e n t i c a l with = C^(r)... t h i s i s in general not so... Function (5 . 5 . 7J. . . overestimates^ at a l l levels of r where the system in use gives a wage curve concave to the o r i g i n . It underestimates \ when the wage curve is convex. It only gives the correct <\ when the wage curve is a straight l i n e . . . and a surrogate production function exists i f and only i f a l l wage curves are straight l i n e s . But (and this i s the big but) in the case of a straight l i n e envelope We should then have to admit that the "marginal products" [w & r ] change when the r a t i o of c a p i t a l to labour does not (Garegnani, 1970) Another way to look at thi s is to write Y = f ( r\ , L ) in intensive form, y = f(K ), i . e . , i t is homogeneous of degree 1. Then, by Euler's theorem we have (Harcourt, 1 9 6 9 ) : = rk + w Taking the t o t a l d i f f e r e n t i a l we have dvj = rdk + K d r + d v Divide through by giving 1 = ijk + kdr + dw E t ceo dy dvj dy ^ D : >- 5 , as expected from the homogeneity degree 1 postulate. In conditions of perfect competition and constant returns 1 38 to scale, neo-classical economics assumes that r = d a , i.e. f = i i substituting t h i s back into... 5.5.8 gives 1 - i + C k d r + d w ) d r However from Euler's theorem which must therefore hold true regardless of any assumptions about the productive process K - — For -pi = W m l ; w f W mfl* - ^  m u s t b e a straight l i n e . Recall the discussion of f i g . 5.1, t a n / i V o f T y v ) r " y ~ ^ . — d^ ihowever dw ~r"^ - ~ i s tangent to the curve of oC at I by d e f i n i t i o n . The two coincide only in the case where the w—Y r e l a t i o n i s a straight l i n e . But the straight l i n e case i s the one where the le v e l of K/L does not alter as W and r a l t e r . Remember also that they must a l t e r in response to changes in r and w i f the value of c a p i t a l in production i s to remain constant. There i s only way in which \J and r can change. Have the r\^J_ r a t i o a l t e r and keep ( W m w -W )/r a straight l i n e . That i s , i f the point ( VJ , T ) appeared at a point in the \3, t plane and the wage curve swung around to meet i t . In thi s case, however the production function 1 39 producing the wage curve i s superfluous, describing a change i t cannot explain (cf. Fisher, 1971). The requirements made of the neo-classical production function, that i t account for changes in output/head and the capital/labour r a t i o are self contradictory f and thi s i s true whether i t i s used in a surrogate or micro—economic context. 5.6 Nuti On Vintage Models And The Value Of Capital Nuti(1970a) has extended t h i s analysis as indicated e a r l i e r , by disaggregating the c a p i t a l good sector in Samuelson's system of price equations and looking at the vintage c a p i t a l defence of neo-classical theory. His is a putty—clay model with the c o e f f i c i e n t of putty output/unit input a- being negative . in those periods in which i t is being invested in the production of clay machinery, and posit i v e when that machinery i s producing putty output. The value of any process i s given by CO. • • i • • -i\_=_time periods. \J = ^ C0Lt — VLL)C1-*- = 0 \TV Ufetime o-f macKiae from Uo the start of its coivstrnctlon) therefore ? n . / i a r r ' w - 1 3 0 .£ a tCi+ r) with wages being paid in terms of putty. He argues that because the neo-classical economists merely look at the vintage once 1 40 they have got hold of i t ( i . e . , examine the change in value of a fixed c o e f f i c i e n t s machine already in production), that in fact the neo-classical vintage model never leaces the confines of the malleable world. The reason why they begin with a machine already in production i s that i t s value i s the c a p i t a l i s e d value of the expected future revenues i t s products w i l l bring i n , not the value of the work that went into producing i t . The neo-classical vintage model therefore makes two i m p l i c i t assumpt ions: 1) that no labour is required in the act of investment, i. e . , that the process of turning putty into clay i s costless; and 2) that there no gestation lags in investment, so that the transformation is instantaneous. If assumption 1 holds, the wage curve given by Nuti w i l l be a straight l i n e . If assumption 2 holds, there can be no change in sign of the f i r s t derivative of the function describing the curve, i . e . , i t i s a necessary condition for the existence of a well behaved production function. Dropping these assumptions means that gross investment i s not simply the amount of labour turned into clay in each period. If investment requires labour, whose wages are paid for out of the putty of the completed machines, then the value of the investment i s the sum of the dated cost of the quantities of labour required in each of the stages of i t s production (however 141 many these might be). The weight of the contributions w i l l be determined by the interest rate, and the time pattern of labour inputs. As wages are paid in putty labour's contribution may also be expressed in terms of dated putty. Gross output w i l l consist of the amount of putty produced in the consumption good sector, i . e . , wages, plus the output of machines: to measure what t h i s w i l l be, we need to know how many stages of production there are in the construction of machinery (equivalent to dif f e r e n t i n d u s t r i a l sectors in the investment good process), so that consumption requirements can be calculated. I f , using Keynesian terminology, we write X = p Q = V J N c + V JN^. Where and H't are employment in the consumption and investment goods sectors respectively, and X equals gross output, then the value of the investment is given by: ^ - V J N c = g N x i . e . , gross surplus output from the consumption good sector is exactly equal to the value of the work done in the investment good sector (Kregel, 1973), and thi s investment can obviously be disaggregated so that X - VJ N r = vi N, + VJ N_ + . . . + V JH, + Mr Net output i s gross output minus depreciation, or in Keynesian terminology again, where £ i s consumption in time t , and CT< i s gross investment 1 42 in time t . Net output in time t i s \ -- = c t + 3 t - j ; . ^ c t + i t where V i s net output and I i s net investment. To measure net output, you need as many other sectors as there as there stages in the l i f e t i m e s of machines, r e c a l l i n g that each machine i s a d i f f e r e n t commodity. Output/head Whether gross or net would then depend both on _ the rate of interest — determining the price of each machine in terms of putty — and the growth rate, determining the r e l a t i v e proportions of putty and machines of a l l kinds in t o t a l output. The assumption of the costless transformation of putty into clay and the use of gross measures evade this fundamental issue of c a p i t a l theory. (Nuti, 1970a) The point of t h i s reference to "gross measures" in this quotation can be seen from Hahn and Matthews (1964) account of vintage models: The emphasis in the vintage approach s h i f t s from net savings and investment to gross savings and investment. Replacement of one machine by another does affect the status quo. Most models assume .a constant r a t i o of gross investment to gross returns. The assumption i s a convenience since the l e v e l of net investment in the sense of net addition to the c a p i t a l stock in v a l u e terms that results from any l e v e l of gross investment is something that be calculated without solving for the whole system, (emphasis added) Nuti uses his model to investigate aspects of technical choice in a c a p i t a l i s t economic and in centralised and decentralised s o c i a l i s t economies. In not one case does the concept of the value of capital/head determine technical choice. Reswitching and capital—reversing occur not only for d i f f e r e n t ranges of value of the interest rate, but also of the growth 1 43 rate. The role of the value of c a p i t a l in a c a p i t a l i s t economy... is only in s e t t l i n g transactions between c a p i t a l i s t firms, to determine the value of the legal right to use machinery and the value of the pieces of paper embodying those r i g h t s . It is necessary to determine d i s t r i b u t i o n of income, not between the haves and the have-nots, but between the haves (Nuti, 1970a) The effect of the use of the concept of the value of capital/head in analysis i s to pass over the relation between the time pattern of labour inputs and the time pattern of outputs into which any output can be resolved and to establish instead a r e l a t i o n between current output and current labour. To th i s purpose the current value of the c a p i t a l stock i s needed; a mythical construction in which the past and the future of the economy are telescoped into the present. Attention i s focussed not on past labour, but on the present value of the embodiment of past labour. (Nuti, op.cit.) Nuti's argument i s that the use of gross measures in neo-classical vintage models allows one to evade the problem of measuring the value of net investment (as indeed Hahn and Matthews state i t does), and thus evade the issue of having to measure the value of c a p i t a l , since then they (neo-classical vintage models) would have e x p l i c i t l y to f a l l back on homogeneous constructs. This i s not an innocent f a i l i n g either, since in focussing on the present value of the embodiment of past labour, i t s current productiveness can be taken to provide a j u s t i f i c a t i o n for the a t t r i b u t i o n of the surplus of current output over the wage b i l l to those who have appropriated the embodiment of past labour, thereby providing the current basis of future 144 appropriation (Nuti, n. d., quoted in Harcourt, 1972) 5.7 Sraffan And Marxist Theories Of Rent We mentioned e a r l i e r that land along with raw materials and labour are treated as non—basics in the Sraffa system. Differences in land qu a l i t y owing to transport costs are not considered, only differences in f e r t i l i t y . Where there are differences in f e r t i l i t y , the production equations for a single crop w i l l be written as follows: + C C i p c + • • + K t i p k ) C i + r ) + L c « +/\p ± Here there are YV q u a l i t i e s of land on which rv processes of producing crop Q take place, y i e l d i n g a set of rt rents A. . The prices (other than p^. ) wages and p r o f i t s are set in the construction of the standard system. For the crop Q, there are, as might be expected, P.+1 unknowns, the price, p c and the r\ rents, , and the system has one degree of freedom. In order to make i t determinate, one of the rents must be zero or 1 45 the relevant solution being always the one in which the p's are £ 0 . S t r i c t l y speaking of course, no rent has to be zero, but one must be set, and since rents arise out of the d i f f e r e n t i a l f e r t i l i t y of the lands, the d i f f e r e n t i a l component of the least productive land in use w i l l necessarily be zero. For convenience therefore, the non d i f f e r e n t i a l component i s assumed zero (but see below). Only the process of production on the no—rent land can enter the standard system i f the crop is a basic good, as non—basics such as land cannot enter into i t s construction. In the case of land a l l being of the same quality , more than one process of production of a crop w i l l be found i f land is scarce. The rent deriving from the use of a process must be the same however, so that the process that yi e l d s the greater output/acre should also do so at a higher cost/unit of output (given the rul i n g levels of p r o f i t s , wages and p r i c e s ) . Both processes would thus have to enter the standard system, "though with c o — e f f i c i e n t s of opposite signs and of such values as would in the aggregate eliminate land from the means of production of that system" (Sraffa, op. c i t . ) . The f i r s t case (n types of land) gives the situation of the extensive margin, the second case (same land of one quality) gives the situation of the intensive marginal. Clearly both are subject to diminishing returns. In the case of the extensive margin, th i s i s evidenced by the decline in rents with drop in f e r t i l i t y . In the case of the intensive margin, by the fact that greater output i s 1 46 associated with higher cost. To continue with the case of the intensive margin, note that the existence side by side of two methods can be regarded as a stage in a pattern of increasing production at diminishing returns. The increase comes about by means of the extension throughout the area of the method that produces the greater output. Once this method has extended to the whole area, rent w i l l r i s e to a point where a t h i r d method (yielding a greater output at a higher/unit cost) can be introduced. In this case output w i l l increase continuously, though methods w i l l change spasmodically. While sc a r c i t y of land thus provides the background from which rent a r i s e s , the only evidence of this is the duality of methods. If there were no sc a r c i t y , only one method, the cheapest, would be used on the land, and there could be no rent. (Sraffa, op.cit.) The implication of this i s that once a method i s achieved which produces as much as, or more than is necessary, duality of methods, scarcity and thus rent disappear. It i s easy to see where Marx would have got his ideas about monopoly and absolute (or in Harvey's (1973) terms, class—monopoly) rent, being as much apart of rent as d i f f e r e n t i a l rent, the lower organic composition of c a p i t a l ( i . e . , the capital/labour ratio) in agriculture and the resistance of landlords to c a p i t a l i s t penetration of agriculture. The idea that rents w i l l r i s e and r i s e and then disappear seems counter—intuitive, especially i f land i s held in private hands. In such a case i t would appear to be to the landlord's advantage to r e s i s t the penetration of 147 c a p i t a l i s t production into agriculture, keeping the capital/labour r a t i o lower in that sector than in industry as a whole. However Howard and King (1975, pp. 140—1) argue that Marx's analysis i s wrong: Marx's theory of rent is f a t a l l y flawed. He seems to have to have interpreted Ricardo's denial of absolute rent as implying that there must exist in c u l t i v a t i o n land on which no rent i s paid (the extensive margin); and this i s empirically f a l s e . But no such assertion i s required. So long as the marginal unit of capital—and—labour y i e l d s no rent, there is s t i l l a p erfectly v a l i d sense in which a l l rent i s d i f f e r e n t i a l rent, and the value of corn i s s t i l l determined solely by the embodied labour requirements at the margin. Marx's apparent neglect of the intensive margin thus destroys his c r i t i q u e of Ricardian rent theory. His own theory i s rather odd: i t implies, for example that the landlords have an interest in preventing the entry of c a p i t a l into agriculture, which i s surely the reverse of the truth... Nor i s Marx's analysis necessary to preserve the coherence of the labour theory of value. He accepted a Ricardian theory of d i f f e r e n t i a l rent, and t h i s in conjunction with the concept of the intensive margin is a l l that is necessary. Land i s scarce, non-reproducible and privately owned; l i k e antiques and old masters, i t w i l l therefore command a price (though of course i t has no value)... Marx considers landed property as one p a r t i c u l a r form of monopoly. Now the prices of monopolised are determined by supply and demand, not by their values, and there i s no reason why the price of land (land and therefore the level of rent) should not be explained in exactly the same way. Rent i s , then, simply a form of monopoly p r o f i t (in orthodox theory, the order of p r i o r i t y i s reversed, and monopoly p r o f i t i s misleadingly seen as a subdivision of rent)... Marx's complex and rather confusing analysis of rent can thus be seen as redundant as well as inadequate. 1 48 In our case, however, i t was the discussion of Sraffa's account of increasing production at the intensive margin that suggested the basis of Marx's thought. The solution to t h i s seeming paradox i s that landlords merely rent out the land, and have no say in the processes of production on that land. Thus c a p i t a l i s t farmers w i l l seek to increase t h e i r p r o f i t s at the landlords expense within the context of the production process. A l t e r n a t i v e l y , we may say that what may be true for landlords as a class is not necessarily true for any individual landlord. The actions of each landlord in seeking to maximise his own rents, by adopting more and productive methods, leads ultimately to the si t u a t i o n where the productivity of the methods employed drives the rents down to zero. On t h i s account, i t may be possible to accept the conclusions of Marx's analysis, that part of the rent w i l l arise out of monopoly control over land, but deny the v a l i d i t y of the analysis of values out of which i t r i s e s . It must be remembered that the neo—Ricardian system of Sraffa consists only of prices; no values are imputed to any component of the system. Marx on the other hand began from an analysis of the values involved; values that could in p r i n c i p l e be translated into p r i c e s . Absolute rent arose because, although the organic composition (constant/variable) of c a p i t a l i s lower in agriculture than in industry, the rate of creation of surplus value (s) i s based only on the variable c a p i t a l (v). Since v i s the same in both agriculture and industry, t h i s implies that the amount of value (c+v+s) i s greater in agriculture than in 1 49 industry. Yet the rate of p r o f i t , giving notional prices, i s the same in both sectors, but i s calculated on c+v only. Thus i f , in the i n d u s t r i a l sector, values equal prices, in the a g r i c u l t u r a l sector, they w i l l exceed notional prices. However, because of class monopoly, landlords are able to raise. prices of a g r i c u l t u r a l products to their values. The extra revenue received goes to the landlords as absolute rent. In this way, the conditions of creation of surplus value in agriculture account for the phenomena of p r o f i t s and rent and the d i v i s i o n of surplus value between the two. Yet, as Harvey (1973) argues, i f landlords can exercise their monopoly power to raise their prices to where they equal the value of their products, why should they stop there? "Marx poses this question but does not provide a s a t i s f a c t o r y answer." (Harvey, op.cit.) Howard and King state simply that t h i s pre—occupation with values i s unnecessary. Nevertheless this does not v i t i a t e the claim that landlords w i l l exercise monopoly power to extract a levy on land use. However Sraffa's description of the processes occurring at the intensive margin suggests further evaluation may be necessary. Returning to Sraffa's discussion of the situations in which rents have to be determined, two other cases a r i s e . One i s the case of a number of d i f f e r e n t q u a l i t i e s of land, one of which i s devoted exclusively to one crop. The second i s where there are a number of d i f f e r e n t q u a l i t i e s of land (n) and a number of crops (m). Each of the lands grows some but not a l l of the crops, and 1 50 each crop appears on some, but not a l l of the lands. However the overlaps are s u f f i c i e n t l y strong that no one rent can be determined independently of the others. In the f i r s t instance, the land that i s completely specialised becomes an example of intensive production, and i t s rent determined independently of the others. In the second instance, rent i s determinate, i f the number of processes in the a g r i c u l t u r a l sector equals n+m, the number of lands plus the number of crops. This implies that there be at least one land with only one crop on i t . Consider l a ads > / X X c r 0 I * X X i . e . , an n x m matrix of crops and lands, (n=5, m=4) with 5+4=9 entries, no crop appears on a l l 5 rows, nor does any one row' have a l l 4 crops. Row 3 contains just one crop. And no matter how these entries are arranged, there w i l l always be one land which bears only one crop, i f the conditions demanded by Sraffa are observed. If one column was f u l l , rents could be determined for that crop d i r e c t l y , and i t would drop out, becoming a simple case of the extensive margin. If one row was f u l l , then that land would be in scarce supply, and the second case, the intensive margin would come into play d i r e c t l y . The crop grown 151 in row 3 is thus the one grown on the no-rent land in thi s case. Interestingly, Sraffa suggests that Machines of an obsolete type which are s t i l l in use are similar to land, insofar as thay are employed as means of production, though not currently produced. The quasi— rent which i s received for those fixed c a p i t a l items... i s determined in precisely the same way as the rent of land. And l i k e land, such obsolecent instruments have the properties of non—basics and are excluded from the composition of the Standard commodity. However, Sraffa also treats machinery under the heading of fixed c a p i t a l , and gives ways of valuing machinery in production, but these two approaches are not linked. The fact that Sraffa i s "always dealing at an instant of time with those properties of an economic system which are independent of change" (Harcourt, 1969), means that a discussion of a vintage model which i s predicated on technical change i s outside his terms of reference. However, the most interesting fact about Sraffa's model i s that though the processes of production are of constant returns to scale, nonetheless variable returns can be accommodated within i t , so long as these occur in the non—basic components of the system. 1 52 6 NEO—RICARDIAN APPROACHES TO URBAN ANALYSIS 6.1 Von Thunen — Based Land—Use Models The analysis of rent in Sraffa's system abstracts from s p a t i a l considerations, but a clue as to how these might be introduced i s given by Sraffa in his discussion of the order of f e r t i l i t y of the various lands: Which do not exist independently of the rents; that order, as well as the magnitude of the rents themselves, may vary with the variation of and If some kind of s p a t i a l ordering of the magnitudes of the rents can be achieved, then the orders of f e r t i l i t y of the lands w i l l s i m i l a r i l y be determined. The obvious source of such a s p a t i a l ordering i s the von Thunen model. Scott (1975, 1976) has examined how these two approaches may be combined. By a convenient coincidence, Beckmann (1972) has provided an account of a von Thunen model treated as a neo-classical land—use model, allowing easy comparison between the style of the two approaches. 1 53 Scott's 1975 paper opens with an account of a simple neo-classical land—use model as follows: Assume an isotropic plane surrounding a central c i t y , and two crops ( CI , t> ) produced under conditions of perfect competition, and d i f f e r e n t i a t e d only by the extent to which transport costs enter into their cost structures, given by D i . ( c j ) p f c l ( i - 0 - b ) . X)^(S) represents the amount of physical transport needed to carry one unit of commodity over the distance 8 (e.g., ton-miles) and p^ is the price of one such unit. Total land rent at distance &, (1 £,(b) , i s given by where ^ is the (assumed to be constant) y i e l d (output/unit area) of a crop, r- i s d i f f e r e n t i a l rent/unit output at &, and \ L is a s p a t i a l l y constant land levy/unit area, Market price/unit p of each crop i s given by where = output - levy/unit of output, and K L i s production costs and i t i s assumed ^ ^ j , ' T n u s the price i s the sum of production costs, transport costs and rents. D i f f e r e n t i a l rent i s then and varies with distance only as varies with distance. At the margin of c u l t i v a t i o n of each crop, d i f f e r e n t i a l rent w i l l be zero so that 1 54 r aC«0 = 0 rb<£) = 0 where oC, CJ are the r a d i i of the outer l i m i t s of c u l t i v a t i o n of each crop CL, h respectively. This implies that prices are ultimately limited by marginal conditions so that Moreover, subtracting t h i s expression from the one given for above (6.1.1), gives where ^ =. l£> At any annulus at distance & from the centre where Q^ jj w i l l occupy exclusively a l l the land contained within that annulus. The land levy paid by producers of crop fa i s * b = V a c ^ + V &r 6.1.3 i . e . , i s equal to the d i f f e r e n t i a l rent paid by crop o. at the point where plus the land levy placed on the production of crop CL, given the assumption that VJ^ I^ b " ^ Kfl,- 0 • Suppose now that there i s given a general demand function for each crop f i C ^ t • Supply i s given by the area under c u l t i v a t i o n . Equilibrium requires the equality of supply and demand so that i l t f t ) ^ J a * = ° Ecy* 6.1-fc and There are 3x2 = 6 unknowns p , A L , ~ &,b) and ^ ( ^ C l ^ a n d 1 55 (1X2)+3=5 equations to give them(6.1.2,3,4 , 5 respectively). To make this system determinate, one unknown must be set. Two alternative p o s s i b i l i t i e s are ava i l a b l e . If land i s unlimited, A ^ w i l l be zero because of competition. If land i s limited, then o t w i l l be the outer l i m i t of c u l t i v a t i o n , and X^ is thus a rent derived from the overall s c a r c i t y of land. Scott discusses the r e l a t i o n between the concept represented by X& and Marx's concept of absolute rent. He repeats the c r i t i c i s m s made of Marx's concept, discussed above, adding that i f the landlords did have the power to intercept the entire between price and value, the whole process of equalisation of p r o f i t between sectors of production would be postponed, and the rate of p r o f i t would be driven to zero in a process of recursive re—adjustment and a l l of the surplus value would disappear in the form of absolute rent. Scott concludes his outline of t h i s model with the comment that: The main d i f f i c u l t y with t h i s rather seductively elegant mathematical model [ i s ] that the conditions of production are e n t i r e l y concealed by the disembodied constant production costs, r^o. and K b , so that the detailed e f f e cts of the production process on prices are not discernable F i n a l l y the model f a i l s to situate i t s e l f within any wider and meaningful s o c i o - p o l i t i c a l structure. (Scott, op.cit., p. 10) For reference, the graph of land rent, production costs and transport costs yielded by t h i s model i s shown in f i g . 6.1. Scott introduces his Sraffa based model with a discussion of the FIGURE 6.1: Land Rents / Unit Area and Land Use on the Uniform P l a i n , Assuming v^K^ =. |jb^b 157 Sraffa system. He introduces a composite commodity sector, consisting of a l l i n d u s t r i a l products, and including marketing services for a g r i c u l t u r a l production. It i s assumed not to consume any land, even though i t s employees do. Thus there are three sectors in the model he wishes to consider, crop OL, crop b and good c • After the introduction of the non s p a t i a l model comes the introduction of land, distance, rent and landlords, "who find themselves in an uneasy a l l i a n c e with c a p i t a l i s t s , through the i n s t i t u t i o n of private property." He then writes the equations of t h i s land using system as follows: ^ c P a + \ P b + C c P c K ^ O + k v . + T cp t 4 "r\ * Cp c \ p t = 1 "T = the output of the transport sector in terms of standard physical units, Tg_ and represent transport inputs to the a g r i c u l t u r a l sectors a and b, p^  i s the price of one transport unit, and "R^ and are the t o t a l land rents paid by sectors CL and b . ~l^p t and "T^ pt are also given by The expression "Tcpt has no direc t meaning, i t i s a derived price of the costs of transport of workers in sector C. "It i s assumed that the transport sector pays no rent and does not d i r e c t l y 1 58 consume any of i t s own output" (Scott, op.cit.) also "transport costs are set aside from the p r o f i t function that they too are paid at the end of the production period also, l i k e wages". (Scott, op.cit., p. 15). However th i s represents an error on Scott's part as to the wage fund theory of c a p i t a l . The wage fund i s renewed at the end of every production period, and used to pay wages throughout the next period. There i s no necessity therefore to keep payment of transport costs back to the end of the production period. This i s a serious shortcoming moreover, since expressing transport costs in this way implies that transport i s a non basic in the system, playing a similar, though autonomous role to that of land, that i s to say, transport costs w i l l be a residual in the productive process. This i s not however the worst of the matter. Demand equations are now brought in, in order that t o t a l output might be determined: At equilibrium, then, t o t a l production of each sector of production w i l l disaggregate without residue into derived demand plus f i n a l demand, that i s I - I a - l b - I c - f i C p t ) s O where 1= fW.T fcf 6.1.8 Aggregate rents are given by where 1,1 * r\b,GL,b 03 before. Scott then considers that It i s now apparent that we have a powerful preliminary mechanism for r e l a t i n g a g r i c u l t u r a l and i n d u s t r i a l production in a s p a t i a l context. B a s i c a l l y , 1 59 th i s mechanism consists of the equations [6.1.6] and J6.1.8]. Given for the moment the quantities R a andT^/ as well as T a and Tb as determined by [6.1.7], and assuming that and £ are perfectly free to vary, t h i s system determines a l l prices, the rate of p r o f i t , a l l intermediate inputs and a l l f i n a l demands, and a l l t o t a l outputs. (Scott, 1975, p. 19) F i r s t l y , as Robinson (1961, quoted above) notes, demand equations are an unnecessary intrusion into t h i s system. Net output must equal labour costs, i . e . , £ [ I - p.- U a P V I b Pb + lcPc><* + r) +TiP e+Rc] = £ l l w If demand equations were introduced into this model, they would only be relevant in the case of non—basics, or luxury items, whose prices are not relevant to the solution of t h i s system. In t h i s l i g h t , supply and demand relations account for the d i s t r i b u t i o n of income, not between the haves and havenots, but between the haves, to quote Nuti one more time. Secondly, i t can be seen from the quotation that Scott makes an error, that of believing that transport costs have been given. In fact they have only been established by means of a tautology, in that i f income at any distance is known, then the cost of transportation can be calculated. However, as i t i s the structure of transportation costs that establishes the l e v e l of income obtained for any crop in any area, both transportation costs and incomes are defined in terms of each other. As the s p a t i a l outcomes of the model depend on the d i f f e r e n t i a l rents, and no mechanism for establishing these is given, we s h a l l pass immediately onto Scott's second model. This model begins with a discussion of von Thunen's work 160 and i t s r e l a t i o n to Sraffa's: "The two dimensions of the economic system represented on the one hand by von Thunen, and on the other by Sraffa are i n t r i n s i c a l l y and intimately interdependent." In t h i s version, the price equations for an individual farm are given by t \ P b + baP*b + caff) Cl+p) + law - ^Pa (abpr+ bbPb + Cbp*)Cl + P) + lbW ^ vjbpb Ca tp* + b cp* c + C b P c) Cl+p) + ttVJ = y £ p c Total output in any area i s given by I = TC CU.^*—^^^StPi. , where U-V i s the interval of distance over which the crop is cul t i v a t e d , where pt is the delivered price of input I into a process, and i s given by pf = p. + t L(£ ) £ = <*j p, o - ma^mi <*cu^ «*i<»n c*x,ps • This i s c l e a r l y an improvement, since transport costs are now endogeneous to the production process. However, note the absence of land from the equations. This i s because Scott prefers to treat of rent as excess p r o f i t . Excess p r o f i t i s by d e f i n i t i o n t o t a l revenue minus t o t a l costs, and where costs include a normal p r o f i t . Therefore equally by d e f i n i t i o n , land rent qua excess p r o f i t can only be imputed to that part of t o t a l production that passes through the market. (Scott, 1976) Once again, Scott wants to use demand equations to determine the t o t a l of f i n a l output 161 where 1 ^  gives the t o t a l numbers of farms of type i , \-being the (unit) area of a farm of type L. In addition, supply is a function of available area multiplied by the (constant) yield/area given by Rent i s yielded by transport costs and by s c a r c i t y . Transport costs are responsible for d i f f e r e n t i a l rents, V".(o>): and the land rents/unit area on each side of a switchpoint w i l l be the same, namely: t The market price on the edge of c u l t i v a t i o n at Y sets the base le v e l p r i c e . There w i l l also be a scarcity component to rent, S i ; on each type of land: The l e v e l of S is determined by V , and would be zerotfXis p e r f e c t l y variable so that again either S is fixed or ^ i s fixed, with either variable becoming endogeneous in either case. F i n a l market price can thus be decomposed into p* = [\+t t«> + n ( G W S i ^ ^ r W ^ O 1 62 In t h i s model, there are twenty variables: pft, , p , p* , p* , £., p , A , B, C , rft(oQ, rb(#, g , , S b , 5 t , <X , (i , ^ , and nineteen equations. However, since either § or % can be fixed, the system is determinate. The f i n a l outcome is given in f i g . 6.2. In fact, [ f i g . 6.2] f i n a l l y c l a r i f i e s the role and function of land rent as a sorter and arranger of land uses, for each individual location within the geographical i n t e r v a l [0, ^  ] i s c l e a r l y uniquely occupied by the crop that can pay the highest aggregate rent at that location. (Scott, i b i d . ) Scott then proposes to "embed" within this model" a simple but very general dynamic mechanism... This mechanism i s precisely the process of overall land use i n t e n s i f i c a t i o n . " However, land use i n t e n s i f i c a t i o n turns out to be no more than increasing production at the intensive margin as described by Sraffa. For comparison with the neo-classical approach before making an evaluation of Scott's work, we turn to Beckmann, (1972). Beckmann follows a procedure similar to that of Wicksteed: Although von Thunen was very s p e c i f i c about the use of c a p i t a l in a g r i c u l t u r a l production, to simplify the analysis, we s h a l l assume only two factors of production, land and labour and also assume constant returns to scale. It i s well known that the production function can then be formulated as OuApat _ JI Labour \ (\ c r e. ' ™^ Acre. I Or vj= Q(x) , where |(o)^0 ( 0'CX)>O; (j)" <<0. O-tvd 0-X#'> 0 and 0(%) c production function, - marginal productivity of FIGURE 6.2: Land Rents, Price And Transport Costs On The Uniform Plain (Assuming That 1 64 labour and 0-X0= marginal product of land, X= labour/acre Consider f i r s t the case of a single a g r i c u l t u r a l commodity. At distance & from the c i t y , employment of X workers/acre y i e l d s a p r o f i t / a c r e of g(h,x) = (p - fcb) a.^x-wx-where P = market price, fc = transport costs/ton—mile, 6 i s distance. Vi i s the wage rate, ass.ume.d_equal to $ , scale parameter P r o f i t maximisation implies that | | = 0 = CL(p-tr)0fbo - w, ot = a - ^ r ^ - ^ ^ 6.1A Implicit d i f f e r e n t i a t i o n of 6.1.9 yiel d s where '> Q and 9>x^0 by the second order condition for p r o f i t maximisation. These results are independent of the production function chosen. For two crops, with production functions d^Cx^") r and d i f f e r i n g only in value of the scale parameter , prices p^ and transport costs t^, p r o f i t / a c r e i s then maximisation with respect to gives t P k - t ^ a ^ ' C x , , ) - w = 0 and x k = a k ? j * — - X , In competing for land use, only that product emerges successfully which yie l d s the highest rent/acre. It follows that at a given distance from the market only one a g r i c u l t u r a l 1 65 product w i l l be produced. Moreover each product can only appear over one range of distance, namely at 6 where The order in which the goods are produced away from the market is given by i . e . , the output/acre with one unit of labour times transport costs i s the deciding factor in how the rings are ordered. The remainder of The model consists i n . . . demonstrating the following general propositions (1) that labour input is a continuous function of distance, (2) that land use zones are arranged in the order of decreasing transport costs in addition to the fact that the labour-land r a t i o i s a decreasing function of distance (Renaud, 1972). Renaud however points an inconsistency in Beckmann's model. Beckmann writes the rent function as However in doing so, he has compared a value 9jk^r^ with a physical quantity, the marginal physical product of land (MPPL ) the correct equation should be giving the value of the marginal product of land (VMPL). This, however, does not a f f e c t the rest of the model. Beckmann concludes with a joint discussion of joint production, giving as an example crop rotation, When joint production permits continuously 1 66 variable proportions of outputs, then in general, the more transportable output i s constantly substituted for the less transportable one with increasing distance. Discrete zones occur only for competing, not for combined a c t i v i t i e s . (Beckmann, 1972) The results of Beckmann's model, and of Scott's model, and of Scott's own neo-classical model are very similar to Scott's Sraffa—based models, disappointingly so when we consider the potential that a neo—Ricardian must have for models of rural and urban land use. To understand why t h i s is so we must take a closer look at Scott's model, since i t deviates from the purpose as well as the logic of the theory from which i t i s derived. To begin with, in Scott's second model, there i s no transport sector, so that an analysis of costs in t h i s sector and how they affe c t input prices, i s not possible. This means that despite his r e l a b e l l i n g of the price c o — e f f i c i e n t s with an asterisk to indicate that they are c . i . f . and not f.o.b. prices, transport s t i l l remains exogeneous to the process, and however the costs are set, so also must the s p a t i a l outcome of the model must be. Secondly, there i s no land in the model, because a l l the equations are set for a unit area, and rent is treated as excess p r o f i t . However th i s account of the determination of rent has serious consequences for the purpose of the analysis, since i t only derives from that part of the product which i s required for sale. Thus in contradiction to the c l a s s i c a l s p i r i t of the analysis, we have a value, rent, deriving from exchange, rather than production. To elaborate t h i s point, r e c a l l that we are dealing here with an a g r i c u l t u r a l system, and using an analysis 1 67 derived from the idea of the production of commodities, by means of commodities. It i s essential to such a system that there be at least one basic production process and good, and i t i s d i f f i c u l t to see how the price of milk, say, affects the production of corn other than by a f f e c t i n g the competition for land. One commodity that is basic to any space economy i s transportation, but t h i s of course is not considered in Scott's (second) model. It i s not possible for these reasons to construct a standard system in Scott's model from his i n i t i a l equations, so that the maximum rate of p r o f i t , and t o t a l surplus product cannot be calculated. This is why demand equations must be brought i n . F i n a l l y , Scott's analysis of market prices i s t a u t o l o g i c a l , for the same reason that his analysis of transport costs i s inadequate, c . i . f . prices are the sum of f.o.b. prices plus transport costs plus rents: nowhere are they a function of the wage—profit r e l a t i o n . f.o.b. prices cannot be derived from that r e l a t i o n ; and transport costs and boundary d i f f e r e n t i a l rents are given exogeneously. Comparing the outcomes of Scott's neo-classical model, as given by f i g . 6.1, and his Sraffan model as given by f i g . 6.2 does not reveal any substantial differences, i f we ignore in f i g . 6.2 areas above the rent lines i . e . , the transport costs, which are merely the dual of the d i f f e r e n t i a l rents. ^ ,~K^ now become S a , s b r S« and k (_ (yi e l d times production costs of i ) i s no d i f f e r e n t from pt , the difference being one of scale only. Scott does not make use of the f u l l potential of the Sraffa model and as a result ; ' v > f . > ; i s 168 forced to make some unnecessary assumptions which lead to serious flaws in the development and solution of the model. 6.2 Reswitching P o s s i b i l i t i e s In Neo-Ricardian Land-Use Models Another attempt at a Sraffan land use model i s given by Steedman and Metcalfe "Reswitching and Primary Input Use" (1972). In thi s model, they are interested in how varying land/labour ratios a f f e c t choice of technique as the p r o f i t rate var i e s . Each technique, which is considered to require two sectors, plus inputs of land and labour, can be expressed in terms of an input—output matrix as follows: I n P a t s 1 2 labour Land a (X-ft* / where column 1 gives inputs of commodities 1 and 2, labour and land required to produce one unit of gross output of commodity 1, 2. The price equations are 169 Pi. = Cttixft, + a i X p i)Ci+ T) + wa.x..+WI\a. where w,W represent the cost of labour and land respectively. Relative prices are given by where V- , Z=^j i . e . , rents/wages It can be shown that 6p/6i> 0. However the response of p to changes in r i s unspecifiable in advance, they may r i s e , f a l l or remain unchanged, for any given l e v e l of i . w can be expressed as a linear term in V and w i l l vary with W as i l l u s t r a t e d in f i g . 6.3. Note that the trade off w i l l change slope as the interest rate changes, i f one commodity i s more or less . land intensive than the other. The wage—profit r e l a t i o n i s non—linear as is well established by now, as also i s the rent— p r o f i t r e l a t i o n , since i t exists in the same formal r e l a t i o n , i l l u s t r a t e d in f i g . 6.4, to the conditions of production as does labour, i l l u s t r a t e d in f i g . 6.5. "R i s the technologically determined rate of surplus. Combining these three relations y i e l d s a composite graph, f i g . 6..-. For any r t h i s shows the corresponding w and V , as ~j varies from o-»oo , and for any ^ , i t shows the variation of wages and rents with the rate of p r o f i t , i . e . , with l = x * , the variation of w and V with r i s traced out by the l i n e Ry*- . Where two techniques (V and b are available, a combined graph of two sets of feasible wage—rent—prof i t relations might be drawn as i l l u s t r a t e d in f i g . 6.^ ;. If we consider now the two 170 EIGURE 6,.3: Behaviour, of Wage - Rent F r o n t i e r as z Changes from 0 - oO at D i f f e r e n t Levels of the I n t e r e s t Rate. ( a f t e r Steadman and Metcalfe,1972) 171 FIGURE 6.6: Composite Graph o f Wage-R e n t - P r o f i t R e l a t i o n s In A Two-Sector Economy, U s i n g Land And Labour In A S i n g l e Technique 172 r 2 = o O < : z = 0 FIGURE 6.8 Wage-Rent-Profit Relations In A Two Sector Two Technique Economy, With Ei t h e r Wages Or Rents Equal To Zero 1 73 cases where i = 0 and z. * oo , we can ' f l a t t e n ' t h i s 3-D graph into two dimensions, corresponding to the planes i and W-0 , •* = 0 respectively, as shown in f i g . 6. /. The wage curves f^b^b an<^ "^Ct^ CL show the variations of the real wage with the p r o f i t rate when rents are zero, and rent—profit curves r^b^b a n ( ^ ^a^O. s i m i l a r i l y show the variation of rents with the p r o f i t rate when wages are zero. As the ra t i o of wage costs to land rentals (the measure of the r e l a t i v e factor i n t e n s i t i e s of the two techniques) varies from z. =0 to z = o O , there w i l l be a switch in technique, but whether that technique is more or less factor intensive in the increasing variable (here land) depends on the l e v e l of the rate of interest. Assume CI is more land intensive than b , then for the various ranges of the rates of interest graphed in f i g . 6. the table gives the dir e c t i o n of the switch as 0 l - > oo in f i g . 6.8. Thus i t can be seen that a switch from a more to a less land intensive method of production as the r a t i o of land to labour costs r i s e s i s not simply a function of the rent/wage r a t i o but also of the p r o f i t rate. This also means that the r e l a t i v e supply of the two commodities (from the two sectors) is not a simple function of their r e l a t i v e price l e v e l s , but also depends on the p r o f i t rate. Steedman and Metcalfe generalise this result to a multi technique s i t u a t i o n . One thing however remains constant that, the r e l a t i o n between wages and rents being l i n e a r , there can only be one switch of technique along the wage—rent f r o n t i e r . 174 TABLE I D i r e c t i o n Of Switch In Technique As0-*7.-»oo , For Various Levels Of r-o i K l i . .. a. t Normal Switch r 1<r<r 1 _ No Switch z,< x < r 3 •b—^ 1 Reverse Switch a.—> a. No Switch 1 75 There may however be more than one switch along the rent—pr o f i t f r o n t i e r , though this is not shown in f i g s . 6.7 or 6.8. This is stated without proof by Steedman and Metcalfe, but i t i s clear from the form of the price equations that t h i s i s so i f the wage—profit relation y i e l d s reswitching relations that the rent—profit relation may do so also. The reason why the movement in factor i n t e n s i t i e s may be the reverse of their associated price ratios i s that The choice between two techniques, a and b at a positi v e rate of p r o f i t i s equivalent to the choice between the two imaginary techniques a and b at a zero rate, of p r o f i t when each commodity's input—output c o — e f f i c i e n t s for the imaginary technique are equal to (1+r) times the corresponding c o — e f f i c i e n t for the actual technique, and the corresponding d i r e c t land/labour ratios are_ the same. Total land/labour i n t e n s i t i e s for a and b w i l l , in general, depend on r ; even though a i s more land intensive than b^ A m&vj f o r some r be less land intensive than. b . fVs the rent-w&Qe rat io i s notioncvl\.vj increased, o\e swUcK vliU aWia\js be to the 1*55 land Uvten-sive urxa^lharij technique, but m&j^ b'e to the m o r e or l&ss Land i n -tensive actual' .technique, ( i b i d . , footnote) F i g . 6.9 shows how these results translate into the multi technique context, for positive values of T, the two land intensity curves are no longer monotonically decreasing functions of the rent—wage r a t i o . Consequently for several d i f f e r e n t values of z, here z ' , t" , z" ' , the chosen technique involves production of one of the commodities with the same t o t a l primary input proportions. Although the curves must have maxima and minima at the same values of x , they must otherwise be of d i f f e r e n t shape, otherwise reswitching could occur between them as z varied, "contradicting the fact that techniques never FIGURE' 6.9: Wage-Rent-Profit Relations In A Two Sector, Multi-Technique Economy 177 reswitch with reference to variations in 7.." (Steedman and Metcalfe, op.cit.) This model i s of some relevance to a s p a t i a l analysis o£> urban or rural land use. In the r u r a l case, the cost of increasing land intensity w i l l be a d i r e c t function of transport costs, and not only as they a f f e c t the production process. Nevertheless the increase inW along i t s axis w i l l also express some function of distance, so that the p r o f i t rate w i l l vary, declining from a maximum obtainable when to zero which gives A J m a x for a given wage rate. In the rural sector, employment is assumed diffused so that there i s no s p a t i a l variation in wage rates. Nonetheless reswitching cannot be ruled out, between less and more land intensive techniques, as the r/vJ r a t i o f a l l s with distance. However t h i s i s less l i k e l y to occur in a g r i c u l t u r a l production than urban for reasons which w i l l become clear once the urban case has been reviewed. For the urban sit u a t i o n , r e c a l l that multiplying the input—output c o — e f f i c i e n t s of the real technique by (1+fP ) gave the c o — e f f i c i e n t s of the imaginary technique and that t h i s could be used to explain the pattern of reswitching. Clearly, t h i s w i l l only be the case i f there is no j o i n t production, i . e . , no fixed c a p i t a l , otherwise from the outline of Nuti's argument the 0- - I m u l t i p l i e r to use w i l l be 3L(l-*r} , where ^  is the length of l i f e I of the fixed c a p i t a l . For an urban entreprise, inputs w i l l be both time dated and s p a t i a l l y s i g n i f i e d . The proportions of land to embodied labour 178 in each input w i l l vary considerably, (so also, therefore, w i l l the proportion of land to dir e c t labour, destroying the l i n e a r i t y of t h i s relation) probably more so than a g r i c u l t u r a l production, consequently s p a t i a l reswitching i s more feasible in i n d u s t r i a l than in a g r i c u l t u r a l land use. A direct s p a t i a l analogy may be made from Dobb's explanation of reswitching to imagine two alternative processes, one requiring a lot of inputs at recent dates, and a few at considerably e a r l i e r dates, and the other requiring an intermediate number of inputs at intermediate dates. Imagine instead a commodity which requires either a lot of inputs from close by, and a few from further a f i e l d , or an intermediate amount from intermediate sources. If such a commodity were housing services, then c l e a r l y , at high wages and/or transport costs, the f i r s t process would be preferred. At intermediate leve l s of wage and transport costs, i t would be the second alte r n a t i v e which be preferred. As a l l three alternatives can exist at the same time, in space, an approximate account may be given of r e s i d e n t i a l location patterns with luxury apartments and country houses separated by suburban v i l l a s . However the e f f e c t s of d i f f e r e n t time patterns of inputs must also be taken into account, and when thi s is done, the possible permutations w i l l increase geometrically. As we have seen, in the Sraffa model, t h i s dual input l a b e l l i n g is handled by the system of j o i n t production, where every i n d u s t r i a l process involving the same product, but 1 79 produced at di f f e r e n t locations are treated as separate processes. The problem with t h i s construction is that l i k e Scott's attempts, they must remain purely formal and tautologous u n t i l some theory of space can be found to give an account of the ordering of these lands (via the cost of bringing them into production), which can be integrated with Sraffa's approach. This w i l l not be easy, since Sraffa's treatment of the question of land excludes simultaneous treatment of intensive and extensive uses of land to—gether, which i s the basis of von Thunen's approach. One possible solution i s to follow Scott's approach, treating the d i v i s i o n of t o t a l surplus between rents and p r o f i t s as technologically determined in addition to the quantity of t o t a l surplus i t s e l f . Then the maximum rate of p r o f i t in any area would be equal to the maximum rate where transport costs would be minimised, minus the sum of unit transport costs involved in production, weighted by distance in time and space giving r = [R-~T](jl-w) , where ~T = T(distance/unit, dates of delivery, costs/ton mile). Then any production equation would be In the case of an a g r i c u l t u r a l economy with c i r c u l a t i n g c a p i t a l only, t =t(distances/unit, dates of delivery, costs/ton mile, in process c-v ) w i l l obviously be a simple function of distance from the market place, and would y i e l d a von Thunen solution. If there i s fixed c a p i t a l , but only one period of production, s p a t i a l reswitching may occur even i f t c. remains a 180 function of distance from marketplace only. If we consider dir e c t transfers from one firm to another, and multiple periods of production, i . e . , the general case, then further disaggregation of the production equations w i l l be required to account for these additional p o s s i b i l i t i e s . The construction of the Standard System would use that equation in which d i f f e r e n t i a l land land rents equalled zero, i . e . , in which t was a maximum. This is a similar procedure to that of von Thunen's own analysis (Harvey, 1981), giving the rate of p r o f i t when a l l locational advantages have been denied; i t i s therefore the rate for which technology alone is responsible. In this treatment therefore a l l processes are treated as f.o.b., and the conversion to c . i . f . takes place through the application of the function t to the rate of p r o f i t . 181 6.3 Conclusion The usefulness of the Sraffa approach in the analysis of an actual economy i s limited. It consists in pointing out what must be t e c h n i c a l l y and economically feasible on stated assumptions, mainly in order to deny the v a l i d i t y of the neo-classical approach (though in t h i s , i t i s most successful). These li m i t a t i o n s are even clearer when i t is generalised into a s p a t i a l context. Its primary function remains the accounting for the d i v i s i o n of the surplus between wages and p r o f i t s . However, i t i s able to provide an account of why the appearance of certain types of land use at more than one distance from the CBD i s not inconsistent with economic r a t i o n a l i t y . In t h i s regard, i t can point up the fact that the Alonso model of urban land use, consisting as i t does of straight l i n e bid rent curves, can only apply in the case of production of a set of homogeneous capital—consumer goods similar to Samuelson's wage curves. In fact, were Alonso's bid rent curves to represent integrated capital—consumer good industries, similar to Garegnani's treatment, they would have no functional relationship with one another, as they would supply their own needs and create their own demand. Rather than giving a set of competitive bid rents operating in a p a r t i c u l a r urban land market, the Alonso model provides a comparison of three d i f f e r e n t urban economies, characterised by very circumscribed processes of production (constant capital/labour ratios in a l l sectors of production, as with Samuelson), under three d i f f e r e n t dominant technologies 182 (commercial, i n d u s t r i a l and housing). The fact that they are treated as integrated capital—consumer good industries means that they cannot by d e f i n i t i o n have any r e l a t i o n with each other. If they are not in fact to be regarded as integrated capital—consumer good industries, then they must be one sector homogeneous c a p i t a l industries, and their irrelevance to actual urban analysis i s then immediate. The Sraffan analysis t e l l s us that in the Alonso model therefore, either a l l goods must be produced by one stage processes, or that the structure of transportation costs must be the same in a l l stages of production, otherwise the bid rent curves would have to be c u r v i l i n e a r in the same way that Garegnani and Nuti explain that the wage curves must. However the most serious problem with this approach i s that of the assumption of perfect competition. Whereas in a space economy, competition is inevitably monopolistic, and so i s better treated through the application of game theory as for example, in Capozza and Van Order (1978), where variations in the conjectural variation of firms' estimates about competitors' reactions y i e l d results quite in c o n f l i c t with those predicted by non s p a t i a l price theory. For example, exit of firms w i l l be associated with a r i s e in price, so that an increase in prices with no change in demand i s associated with a greater quantity of output per firm. Neo-classical price theory has a firm's output f a l l i n g i f i t s prices r i s e under perfect competition and i n e l a s t i c demand. This phenomenon however i s consistent with the 183 Keynes—Kalecki or post-Keynesian theories of p r i c i n g and output alluded to e a r l i e r , which are, that prices generate revenue to maintain output at a given l e v e l ; i f i t is desired to increase output, prices may have to be increased in order to generate the revenue for the extra output that i s required. An account of post-Keynesian economics and i t s a p p l i c a b i l i t y to urban growth and development is therefore the l o g i c a l conclusion to t h i s review of economic models of urban processes. In concluding t h i s review on the issue of monopoly on the a p p l i c a b i l i t y of the Sraffa system to urban analysis, note that t h i s affects neo-classical economics far more seriously. Spatial monopolies arise out of economies of scale, coupled with transport costs. However, as Sraffa himself argued in 1926, the existence of scale economies ensures the dominance of the conditions of supply and this is in keeping with the s p i r i t of c l a s s i c a l , rather than neo-classical analysis. 1 84 7 POST-KEYNESIAN APPROACHES TO URBAN GROWTH In t h i s section, we review the Keynesian and post-Keynesian l i t e r a t u r e on economic growth and development. To date however, thi s work has had l i t t l e impact on urban economics. Therefore the style of the following chapters i s somewhat more tendentious, arguing a case for the a p p l i c a b i l i t y of post-Keynesian concepts to the analysis of urban growth. The basic question post-Keynesian analysis seeks to answer i s , i f the neo—Ricardian c r i t i q u e of neo-classical economics destroys the attempt to derive the rate of p r o f i t from the techniques of production employed, what does determine the rate of p r o f i t , and how might this remain constant? The answer given i s the l e v e l of investment, which however stimulates growth. Post-Keynesian economics therefore derives from the Harrod—Domar model, which sets out the conditions for the maintenance of a constant rate of p r o f i t in the face of increasing output from increased net investment as a result of a positive rate of p r o f i t . If the l e v e l of investment decides p r o f i t s , then these are independent of the money wage rate, and the real wage rate i s also set by the l e v e l of investment. A 'sticky' wage l e v e l i s therefore essential in investment planning, and i s not the cause of imperfections in the labour market. In Keynesian economics, 1 8 5 supply and demand are not equated by price fluctuations, but by changes in the volume of output. Changes in the volume of output are related to changes in investment levels via the m u l t i p l i e r . Post-Keynesian theory however needs to be distinguished from neo—Keynesian, which attempts to incorporate Keynesian theory into neo-classical economics. Government action along 'Keynesian' l i n e s i s supposed to keep output at the f u l l employment le v e l in the short run. A l l the marginal productivity theorems of the neo-classical analysis of d i s t r i b u t i o n and growth can then be assumed to apply in the long run. Post—Keynesians argue that t h i s procedure gives the long run features which cannot be found in the short run. Neo—Keynesian work, though using in many cases the same models and terminology as post-Keynesian, tends to be much more mechanistic than post-Keynesian, projecting trends into the future in a manner quite contrary to the s p i r i t of Keynes' views about the essential i n c a l c u l a b i l i t y of future prospects. Post-Keynesian work however i s much more cautious, arguing only that i f long run steady growth or fluctuations are to occur, then the economic system must have the c h a r a c t e r i s t i c s described by the model in question. The parameters of these models cannot be expected to remain constant in r e a l i t y . The most v o l a t i l e of these parameters i s investor confidence in the future. The causes of business cycles are therefore largely f i n a n c i a l , and should not be regarded as being caused by changes in output. The major d i f f i c u l t y facing post-Keynesian work i s that i t 186 has a l l been developed in a macro—economic fashion. Even when applied in a regional context, models of regional growth in fact do l i t t l e more than examine the problems of a small country with open borders. In these chapters, a way of disaggregating e f f e c t i v e demand i s put forward. Using t h i s method, i t i s possible to show that a housing shortage can be consistent with economic equilibrium. That is to say, i t cannot be blamed on ad hoc interference with the smooth running of the market, such as government interference. Related to this issue, i t can be shown that i f the housing stock i s regarded as malleable, then housing shortages are inevitable as the growth of c i t i e s changes the forces of demand, while the d u r a b i l i t y of the housing stock places constraints on supply. Especially relevant in t h i s context is the s p a t i a l v a r i a t i o n in patterns of employment and r e s i d e n t i a l location as the c i t y grows. F i n a l l y , Keynesian'analysis is consistent with the c r i t i q u e of u t i l i t y theory presented above, and i s consistent with the presence of class stucturation in society. The c r i t i q u e s of u t i l i t y theory, which can be developed into an account of class structuration, includes the Keynesian c r i t i q u e of neo-classical labour economics as a special case. This is possibly i t s most useful p o t e n t i a l , since there is a marked schizophrenia in urban analysis, in having s o c i o l o g i s t s looking at r e s i d e n t i a l location as the result of an a l l o c a t i o n process, either bureaucratic, class based or due to r a c i a l segreation, and having economists 187 looking at the same outcome as a result of revealed preference. Clearly i t must be a result of both processes, but also a result of the constraints placed on lo c a t i o n a l processes by supply shortages. Structural Marxist approaches tend to give the ideological and bureaucratic processes of the state, the same omniscience and omnipotence that neo-classicals give to the market. A balanced account i s sorely needed. We begin with a review of the basis of post-Keynesian economics. 7.1 Keynes, Kalecki And The Foundations Of Post-Keynesian Economics The heritage of Keynes' work has, as we have seen, been claimed by more than one school of economic thought, from Friedmann (Gowland, 1979), to Samuelson (Pen, 1972), to Robinson (Kregel, 1980). The two p r i n c i p a l claimants to the Keynesian throne are the neo—Keynesians and the post—Keynesians. The neo—Keynesians, of whom Samuelson is the most famous figure, attempt to synthesise neo-classical 'micro'—economics with Keynesian 'macro'—economics. The Keynesian macro—economic formulae give the t o t a l volume of output, while the neo-classical production function gives the d i s t r i b u t i o n of that output. Though we have already seen the problems involved in the 188 use of neo-classical production functions, the neo—Keynesians can and do argue that their attempts to synthesise Keynes work with that of neo-classical economics, i s along l i n e s similar to those pursued by Keynes in his General Theory. Keynes regarded i t "as an important point of agreement " with neo-classical theory that the wage i s equal to the marginal productivity of labour ( i . e . , the f i r s t c l a s s i c a l postulate). Furthermore, the General Theory was given that name because Keynes considered i t to include the neo-classical theory as a special case This claim however i s disputed by the post—Keynesians. Robinson (1976) argues that though Keynes made the second c l a s s i c a l postulate (that the money wage i s given by the marginal d i s u t i l i t y of employment), the focus of his attack on neo-classical theory, in the process he also destroyed the f i r s t . As s h a l l be seen below, he showed that the number of jobs available depends on investment behaviour, and that therefore the cost of labour depended investment behaviour, and that therefore the cost of financing that investment, not technical considerations, was the primary factor in determining wage costs. Furthermore, the post-Keynesian development of the general theory is capable of providing a theory of d i s t r i b u t i o n and p r o f i t s i t s e l f , and therefore does not require the imposition of marginal productivity assumptions to complete i t . Post-Keynesian theory derives as much from the work of 189 Kalecki, published more or less contemporaneously with the General Theory (Kalecki, 1971), as from Keynes himself. The two never collaborated however, so that while their basic ideas concerning the operation of the economic system are the same, the influence of their respective i n t e l l e c t u a l backgrounds, Marshall for Keynes, Marx for Kalecki, led to differences of emphasis in the presentation of those ideas. Post-Keynesian theory has been castigated by Roosevelt(1980) for looking at the economic system through the eyes ot the c a p i t a l i s t , despite the post—Keynesians c r i t i c a l evaluation of neo-classical theory as a h i s t o r i c a l and as an attempt to j u s t i f y the status quo. Whether thi s i s generally true or not, Keynes' own work i s c e r t a i n l y open to that interpretation. However, as Kregel (1980) argues, thi s i s to do the General Theory an i n j u s t i c e : In his exposition of the p o s s i b i l i t y of involuntary unemployment, Keynes chose the assumptions that would give the e x i s t i n g theory the strongest possible case, i . e . , f l e x i b l e wages and prices responding to changes in supply and demand but in a setting of actual h i s t o r i c a l time... Keynes was able to show that even when the c l a s s i c a l assumptions were met, f u l l employment equilibrium was not a necessary r e s u l t . At t h i s point, there were two lines that could be followed, one negative and t h e o r e t i c a l , the other positive and pragmatic. Keynes could have concentrated on why the c l a s s i c a l price mechanism did not produce the intended results of f u l l employment of a l l factors. But instead, being a p r a c t i c a l man facing intolerable unemployment, Keynes completely recast economic theory, emphasising the positive aspects of the new approach... But Keynes... nonetheless retained much of the supply and demand framework of Marshall as the micro basis of his theory; t h i s even after he had i m p l i c i t l y proved that the price system in a r e a l i s t i c monetary economy did not operate as assumed in the nonmonetary 190 c l a s s i c a l world of Say's Law and the quantity theory of money. (Kregel, op.cit. ) The problems with Keynes' work l i e in the exposition of his views rather than the views themselves. Keynes' p r i n c i p a l contribution, the theory of e f f e c t i v e demand, and the reasons why d e f i c i e n t e f f e c t i v e demand is not incompatible with long run equilibrium (in an economic, though c e r t a i n l y not in a s o c i a l sense), necessarily imply the existence of a c a p i t a l i s t i n d u s t r i a l i s e d economy (Pasin e t t i , 1974). The reason offered as to why e f f e c t i v e demand may be less than f u l l employment demand, investors' uncertainty about the future, and the inpenetrable nature of that uncertainty, (as Shackle (1972) puts i t , "Time i s a denial of the omnipotence of reason") was not only a major advance in economic theory, but also a provocative and challenging philosophical p o s i t i o n . Keynes spared his readers even in the deliberately provocative General Theory of 1936, the ultimate force of his conclusion, that rat i o n a l conduct i s an i l l u s i o n and unrelated to the r e a l i t i e s of business. That f i n a l smashing of the i d o l was reserved for the last version of the theory of unemployment, the Quarterly Journal reply to his c r i t i c s . . . Thus then our argument passes from the fundamentals of time and being to the p r a c t i c a l conditions of business. Keynes undertook the destruction of the rati o n a l value—theory i d e a l . In order to be understood, indeed to arrive at his goal from his own staring point of adherence to the t r a d i t i o n a l view, he spoke the language of his opponents. The General Theory of Employment, Interest and Money proceeds in terms of functions and regards variables as being in some sense dependent on each other. But thi s dependence of many variables upon each other i s in vain for the defence of the t r a d i t i o n a l standpoint. For one variable i s l e f t unchained. Investment, the flow of orders for (durable) equipment, i s at the mercy, not of other variables, 191 firmly clasping i t s shoulder in a function—grip, but of the ever dissolving and re—appearing w i l l o'the wisp of expectations; of hopes and fears, of surmise and despondency. Investment i s the maverick variable not f u l l y harnessed into the team. (Shackle, 1972, p. 233) Kalecki, on the other hand brings out the contingent nature of the c a p i t a l i s t system more e x p l i c i t l y . Though "Kalecki has the same views as Keynes concerning (1) the wage bargain, (2) the f u t i l i t y of e f f e c t i n g employment through reduction of the money wage, (3) the emphasis on time, and (4) the concern with e f f e c t i v e demand" (Kregel, 1971), his Marxist background leads him to take as read that control of the investment decision by c a p i t a l i s t s can lead to de f i c i e n t aggregate demand, through the creation of a pattern of output which serves their interests rather than that of society as a whole. This treatment of the investment decision in terms of s o c i a l control gives Kalecki's work a richer treatment of the r e l a t i o n of investment to p r i c i n g decisions via the degree of monopoly prevalent in the economy. The determination of the rate of interest however i s dealt with very summarily however. Thus Kalecki goes d i r e c t l y to the r e l a t i o n between the l e v e l of investment, income d i s t r i b u t i o n and business cycles, which was how he f i r s t approached the subject, while Keynes, on the other hand, was as concerned with showing how the f i n a n c i a l considerations surrounding the investment decision could not be captured by the p r e v a i l i n g orthodoxy, which could not therefore give an adequate explanation of the depression, as to show that his theory could. Robinson writes that Keynes could have been saved much trouble 1 92 had he also proceeded from Marx, rather than from Marshall, but i t i s doubtful i f then his ideas would have had so wide an acceptance. By conducting his arguments in the language of the p r e v a i l i n g Marshallian orthodoxy, and demonstrating i t s internal d e f i c i e n c i e s , he was able to force those academics holding the orthodox viewpoint into confronting i t s deficencies and into admitting that the economic system of theory as well as of r e a l i t y was perfectly capable of existing in a permanent state of u n d e r u t i l i s a t i o n of resources, and not least of a l l to demonstrate that this state of a f f a i r s was capable of being r e c t i f i e d . Although Kalecki goes to the heart of the matter more d i r e c t l y than Keynes, he offers no policy p r e s c r i p t i o n s 1 , and i s interested less in the d e f i c i e n c i e s of the orthodox theory than in the d e f i c i e n c i e s of capitalism i t s e l f . Thus Kalecki i s not interested in disproving theories of voluntary unemployment, c f . his famous remark that " i f these results appear paradoxical, i t i s not because of the theory presented, but because ot the paradoxes of the c a p i t a l i s t system i t s e l f . " In the next chapter we s h a l l provide accounts of Keynesian theory. We s h a l l look at the basic short period theory, common to both the post-Keynesian and neo—Keynesian developments of the 'At l e a s t , not in his o r i g i n a l a r t i c l e s , though in 1943, he argued that once governments had the knowledge of how to regulate the economy, a p o l i t i c a l business cycle could be expected. Furthermore, i f government control of investment is to be used to maintain f u l l employment, then the question of the content of that investment becomes the v i t a l issue (Robinson, 1971b). 1 93 the theory, 'macro'—economic models of urban growth and development, which owe more to the neo—Keynesian than the post-Keynesian theory. Ww s h a l l review the implications that a space economy has for economic theory, with p a r t i c u l a r reference however to Keynesian theory. F i n a l l y we demonstrate how post-Keynesian theory may be developed to produce an account of a housing shortage which i s compatible with economic equilibrium. In doing so, we s h a l l also provide a c r i t i q u e of the t h i r d strand of neo-classical economics, namely general equilibrium theory. 7.2 Ef f e c t i v e Demand And Location Theory I The p r i n c i p l e of e f f e c t i v e demand i s the basis of Keynes' analysis of the economic system. It is a very simple idea, but one which i s c r u c i a l to the understanding of an i n d u s t r i a l economy. It i s described by Pasinetti (1974) as follows. Among the p e c u l i a r i t i e s which an i n d u s t r i a l economy has acquired, with respect to more primitive (ag r i c u l t u r a l ) s o c i e t i e s , there i s one that requires us to make a d i s t i n c t i o n between productive capacity and actual production. In primitive ( a g r i c u l t u r a l ) s o c i e t i e s , each farmer t r i e s to produce as much as he can. He w i l l then take whatever amount of his produce w i l l fetch the price the market makes. In an i n d u s t r i a l society i t is not so. At any given point of time, productive capacity i s indeed what i t i s — i t cannot be changed. But productive capacity does not mean p r o d u c t i o n — i t only means potential production. In order that there may be actual production, there 1 94 must be e f f e c t i v e demand... Quite simply, demand generates income. If producers were to expect a f a l l in demand, they would reduce production accordingly, quite irrespective of the level of their productive capacity. And they would do the opposite i f they were to expect an increase .of demand. Therefore as long as thereisidle capacity "to use, fluctuations of demand generate fluctuations of production while prices w i l l remain more or less unaffected... among the factors concurring to determine prices, fluctuations of demand have become unimportant. Therefore the t r a d i t i o n a l response mechanism of price changes having become inoperative, another response mechanism is brought into use. To changes in demand, producers respond by changing production. This has a very serious consequence. Changes in production e n t a i l changes in the u t i l i s a t i o n of existing productive capacity and in the employment of labour. A f a l l of t o t a l demand generates unemployment and a slump — a b i t t e r r e a l i t y so often experienced in c a p i t a l i s t economies. There are machines and workers able to man them, but they a l l remain i d l e for lack of e f f e c t i v e demand. (Pa s i n e t t i , op.cit. ) Pasinetti also makes the point that the gap between potential production and e f f e c t i v e demand w i l l in a l l l i k e l i h o o d be d i f f e r e n t in d i f f e r e n t i n d u s t r i a l sectors. We may also add that i t i s l i k e l y that these differences w i l l be compounded by the existence of d i f f e r i n g regional economic structures. The question of who controls the economic system thus becomes of v i t a l importance, since the system w i l l be run in their interest; even i f this should be the enlightened s e l f — i n t e r e s t of c l a s s i c a l theory, there i s yet no guarantee that a l l interests w i l l be met. To a certain extent t h i s may be said to follow from Sraffa's (1926) demonstration that under increasing returns to 195 scale, conditions of demand are not independent of conditions of supply, and that conditions of supply are l o g i c a l l y prior in the analysis. However i t also follows from the p r i n c i p l e of e f f e c t i v e demand. For the system to remain viable, c a p i t a l i s t s must provide the workers with consumption goods s u f f i c i e n t to keep the d i v i s i o n of output between workers and c a p i t a l i s t s constant, otherwise the system would be precipitated into a slump. Since the c a p i t a l i s t s , who decide the d i s t r i b u t i o n of income, are in charge of production, the conditions of supply are l o g i c a l l y prior as before, but they need not necessarily meet the requirements of demand ( i . e . , demand for employment). So not only are workers' locatio n a l preferences not a determining factor in deciding r e s i d e n t i a l location, but also th e i r requirements (potential demand) need not be a factor in deciding where c a p i t a l i s t s (or a l t e r n a t i v e l y the requirements of the c a p i t a l i s t system) decide they should be located. We need therefore to look at the determinants of investment (which structures the . conditions of supply) in order to discover whether in fact workers' e f f e c t i v e demand for housing is l i k e l y to be s u f f i c i e n t l y strong to effect r e s i d e n t i a l development locati o n . 196 7.3 Characteristics Of Investment I Total demand in the system i s defined to be equal to the t o t a l wages b i l l plus c a p i t a l i s t s ' consumption (Kregel, 1973) : J) = V / N + c P w = f^oneij wages N - number oi workers op ; capitalists* OmsampUo/i (c) oat of profits CP) If we define investment in any period as the work done in the machine sector, and not merely as the value of machines i n s t a l l e d and completed in that period (Nuti, 1970a, Robinson and Eatwell, 1973) we may divide employment into two sectors; investment—providing machine goods, both to reproduce i t s e l f and also for the consumption good s e c t o r — p r o v i d i n g the wage goods. p = wNr + wK, + cP C v In equilibrium, demand equals supply D = pQ. Q - output of the constunption sector, p-. price level Total p r o f i t on the sale of consumption goods equals sales minus the wage b i l l in c P Q = pQ - V J N c = w(M L+ c P Thus t o t a l p r o f i t s on sale of Q i s just equal to the payment of the wages b i l l in the investment sector, and the c a p i t a l i s t s consumption out of p r o f i t s . Note that therefore no matter how large the consumption out of p r o f i t , i t i s always financed by the return on sales of consumption goods. This i s the widows' cruse—which never emptied no matter how much was poured out of 1 97 i t — t h e o r y of d i s t r i b u t i o n (Keynes, 1930). Or as Kalecki i s claimed to have expressed i t , "the workers spend what they get, the c a p i t a l i s t s get what they spend". Kalecki (1933) writes t h i s equation as follows Gross P r o f i t s = Gross Investment + C a p i t a l i s t s Consumpt ion. What is the significance of t h i s equation? Does i t mean that p r o f i t s in a given period determine c a p i t a l i s t s consumption or the reverse of this? The answer to t h i s question depends on which of these items i s d i r e c t l y subject to the decisions of c a p i t a l i s t s . Now i t is clear that c a p i t a l i s t s may decide to consume more or to invest more in a given period, but they cannot decide to earn more. It is therefore their investment and consumption decisions which determine p r o f i t s and not vice versa. I f , for s i m p l i c i t y we assume that c a p i t a l i s t s consumption is a constant proportion of gross p r o f i t s , i . e . , that cP= IZ-s^ P^ , (where S c i s the c a p i t a l i s t s ' propensity to save) we may concentrate on investment in any period as the s i g n i f i c a n t variable. The amount of investment is determined by the rate of p r o f i t expected on projects insofar as they obtain an economic rent with respect to the rate of money interest, i . e . , the opportunity cost of holding money. Money i s to be regarded as having three main uses: as a unit of account for clearing t h i r d party debts, i . e . , as a medium of exchange; as a hedge against unexpected accidents, i . e . , as a precaution against the uncertainties of the future; and a speculative function. The second two uses r e f l e c t the role of money as a store of value, 198 but i t i s the t h i r d use which i s peculiar to a c a p i t a l i s t economy (Keynes,1937). Depending on speculators' views about future prospects- for investment, prospects which are e s s e n t i a l l y incalculable, their desire for money ( l i q u i d i t y preference) w i l l be greater or lesser. The rate of interest thus measures the supply and demand for money. The lower the rate of interest, the more money i s available for investment purposes. This, plus the amount required for transactions determines the t o t a l quantity of money in the economy. The l e v e l of prices with respect to money wages, i . e . , the real wage l e v e l i s determined by the amount of investment ordered, plus c a p i t a l i s t s ' consumption. Returning to our algebraic representation, V = )± - 0 - " cP r P N Assuming v, Q (and therefore also wN. ) fixed in thi s short period, the larger is therefore the lower is ^ , for any given wage rate. As we have made cP - (I~SC)PQ , then the only variable on the right hand side i s N = N + N- . Since N i s assumed fixed, the only variable l e f t to us therefore i s M^, the le v e l of employment in the machine sector, which we may regard as a surrogate for the l e v e l of investment for the time being. Thus we reach the important result that investment i s paid for out of savings. In thi s case the savings are enforced by a f a l l in the real wage l e v e l through in prices. Looking at i t in physical terms, i . e . , the right hand side of the equation, we 199 can see t h a t the l e s s consumption good output t o go round amongst the wo r k e r s , e i t h e r t he h i g h e r the p r o p o r t i o n of c a p i t a l i s t s ' consumption which the workers a r e p a y i n g f o r , or the l a r g e r the amount of r e s o u r c e s which are b e i n g devoted t o in v e s t m e n t . I f t h e r e i s l e s s o u t p u t t o go round per worker, the workers are p e r f o r c e b e i n g made t o save. However, though s a v i n g s thus e q u a l investment i n the end, the amount of s a v i n g s does not determ i n e the amount of i n v e s t m e n t . T h i s may be seen i f we put the above account i n a c a u s a l sequence ( P a s i n e t t i , 1974) 0 ( L,M) I -* H'CE.O - * I 6 f 7.3.1a L — l i q u i d i t y p r e f e r e n c e , f [ — money s u p p l y , I — r a t e of i n t e r e s t , £ — ex p e c t e d p r o f i t a b i l i t y of investment p r o j e c t s , J — i n v e s t m e n t ; (j), a r b i t r a r y f u n c t i o n s . T o t a l income Y» e q u a l s consumption p l u s i n v e s t m e n t , however, consumption depends on the l e v e l of income. Thus I = C + I ^ a 7.3.1 C = K t ) f - propensity to corviame E ^ - J - J 3 OT Q, ~ f\-+cY ~ 0. I mew Ckp^o*'^cH&^ — c ^ rrva^mftv pr&^ eivfttt^ -rb C o n s u l £op~[1.W S u b s t i t u t i n g 3 i n t o 2 and expanding by T a y l o r s e r i e s , (but n e g l e c t i n g terms-of h i g h e r than the f i r s t ) g i v e s 7 / ( 1 - 0 - 1 and and , . . .; - — — — \ l , tt\e itweitment mu-iHpUer Efl^- "[ 3 , 5 T h e r e f o r e w i t h t d e t e r m i n e d as above we may then complete 200 7.3.1a as follows Since savings, S , = ^ f - C r by def i n i t i o n , S = I , in the sense however t h a t ' I - ^ * The l e v e l of savings i s a function of the r e l a t i o n of t o t a l income to consumption, and as this i s determined by the l e v e l of investment, the amount of savings always accommodates i t s e l f ex post to the l e v e l of investment decided ex ante. Thus the rate of interest does not equate savings and investment as i t did in the pre—Keynesian theory, but, as has been seen, the supply and demand for money. This approach to economic analysis, of asymetrical causal ordering, i s one which is opposed to that of neo-classical economics, where the the solution of a system of simultaneous equations can only proceed on the assumption that a l l 'factors' enter independently of each other, and that then each has a dire c t e f f e c t on every other one. Such a procedure i s t y p i c a l of p o s i t i v i s t i c analysis " c h a r a c t e r i s t i c a l l y a h i s t o r i c a l biased to—wards psychologism (at t r i b u t i n g behaviour to innate autonomous desires, needs, demands) and conversely biased against explanation in terms of s o c i a l structure" (Sayer, 1976). Keynes expressly c r i t i c i s e d such a procedure It is a great fault of symbolic pseudo—mathematical methods of formalising a system of economic analysis... that they expressly assume s t r i c t independence between the factors involved and lose a l l their cogency and authority i f t h i s i s not allowed; whereas in ordinary discourse, where we are not b l i n d l y manipulating, but know a l l the time what 201 the words mean, we can keep "at the back of our heads" the necessary reserves and q u a l i f i c a t i o n s and the adjustments which we s h a l l have to make later on, in a way in which we cannot keep complicated p a r t i a l d i f f e r e n t i a l s "at the back" of several pages of algebra which assume they a l l vanish. Too large a proportion of recent "mathematical economics are mere concoctions, as imprecise as the i n i t i a l assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols. (Keynes, 1936, p. 297-8) On the other hand, Keynes outlines his procedure as follows The d i v i s i o n of the determinants of the economic system into two groups of given factors and independent variables i s , of course quite a r b i t r a r y from any absolute standpoint. The d i v i s i o n must be made e n t i r e l y on the basis of experience, so as to correspond on the one hand to factors in which the changes seem to be so slow or so l i t t l e relevant to our quaes itum; and on the other hand to those factors which are found in practice to exercise a dominant influence on our quaes itum. Our present object i s to discover what determines at any time the national income of a given economic system... Which means in a study so complex as that of economics, in which we cannot hope to make completely accurate generalisations, the factors whose changes mainly determine our quaes itum. (Keynes op.cit. P. 247) If i t is objected that there must be some kind of feedback take place in the future course of time. In the present, the only feedback i s a reflexive one, the state of expectations of the prospective yi e l d s of planned investments. from Y and C remember that such feedback can only 202 7.4 Characteristics Of Investment II So far in our examination of investment behaviour and the demand for housing, we have been concerned only with the general si t u a t i o n , not with the conditions peculiar to the urban case. We now turn to an examination of the factors a f f e c t i n g the urban mult i p l i e r . The urban economy is very susceptible to outside influences and a l l kinds of factors can act to augment or depress e f f e c t i v e demand. These are referred to as " i n j e c t i o n s " and "leakages" respectively. Returning to our equation, P N anything which acts to increase the flow of money to investors is an i n j e c t i o n . Investment does th i s by r a i s i n g p r e l a t i v e to V/. Exports do so also by l) obtaining foreign exchange, 2) by reducing the amount of Q l e f t to go round in the domestic economy: prices w i l l therefore r i s e , and entrepreneurs' incomes w i l l therefore also r i s e . Direct government action may also be taken to stimulate the economy. If government action increases N then the output,Q , may r i s e without the entrepreneurs having to bear the cost of the requisite investment, and since p has not risen, without the consumer having to bear the cost through higher prices either. Thus j = l ' + X + G and L = S + M + T 203 Leakages are simply the opposite of injections, reducing the flow of funds to entrepreneurs, depressing investment and thus e f f e c t i v e demand. Saving, i . e . , hoarding, has t h i s e f f e c t . Suppose people decide not to buy as much of Q as previously. In order to clear the market, entrepreneurs must must lower p, net investment and thus N must also drop in order to preserve the equality. Note that t h i s implies that the real wage is a goods market clearing price, not as in pre—Keynesian theory, a labour market clearing price (Solow and S t i g l i t z , 1968), a view which led pre—Keynesian theorists to argue that involuntary unemployment could not e x i s t . This i s the only way in which savings can determine investment, but note that i t i s in the opposite di r e c t i o n to that suggested by the pre—Keynesians (and many modern economists also, e.g., Hicks below). Here an increase in savings leads to a decrease in investment, and so i s analogous to increased l i q u i d i t y preference on the part of speculators. Spending on imports however has exactly the same ef f e c t , less of the domestically produced consumption good w i l l now be purchased, u n t i l i t s i t s price f a l l s to the amount that consumers are w i l l i n g to spend on domestically produced items. F i n a l l y taxation reduces net consumption and thus investment also. Direct taxation does so by reducing the amount of money available to spend on consumption goods, so prices must once again f a l l in order to clear the market. Indirect taxation raises p but the money so raised i s not necessarily used to finance investment ( i t might go in foreign aid for example), and 204 since consumers again find their real income reduced, consumption and hence net investment w i l l f a l l also. Note that throughout th i s discussion, we keep W constant (except as affected by di r e c t taxation). Indeed, i f V I were f l e x i b l e , employment could fluctuate v i o l e n t l y between zero and f u l l employment. As t o t a l wages varied between the amount capable of purchasing a l l the output p o t e n t i a l l y available at f u l l capacity operation of plant, and no wages at a l l . A 'sticky' wage rate i s thus a necessity for confident investment planning. As we have said, both injections and leakages are l i k e l y to be s i g n i f i c a n t in the urban economy. However, there i s l i k e l y to be a difference in the lengths of time which these take to a f f e c t the l e v e l of a c t i v i t y in the urban economy. To account for t h i s , we need to introduce time lags into the m u l t i p l i e r (P a s i n e t t i , 1974). This allows for the effects of investment to be spread over several time periods. If we write • AY = Y n - Y 0 , the t o t a l increase of income from t = 0 to f s f l and C t = ^*c^i-x. i i - e . , consumption which is based on the income of the previous period, a 'lagged consumption function', we obtain A Y . ^ - Y o = K l + c + c l + . . + c r i- 1) or, using the formula for the sum of a geometric progression: x x 1-c Since c<.l, c l e a r l y has a f i n i t e l i m i t as n-^ oo, i . e . , 205 l i m A Y =. - i — I n oo which precisely coincides with expression (7 . 3.5), obtained for the instantaneous m u l t i p l i e r . Thus when consumption decisions l a g behind income, the t o t a l increase in income, given by the m u l t i p l i e r formula l/(l-C) , w i l l be obtained, not immediately, but asymptotically, as time goes by. This process is i l l u s t r a t e d in figure 7.1. At the beginning of the process to—wards the new equilibrium situation there are big leaps — after only 4 or 5 steps, the system i s already near the new equilibrium l e v e l of income, Y . But since each step i s smaller than the previous one, the process slows down as i t goes on. The f i n a l position, though nearly approached after only the f i r s t few steps i s never quite reached exactly. The interesting phenomenon to notice i s that during the whole process, the t o t a l savings that people intend to do or ex ante savings  S(^.x~Ct are d i f f e r e n t from the t o t a l savings they w i l l in fact end up doing, or actual savings YT ~"CT • E x ante savings in f i g . 7.1 are the difference between the 45° l i n e and function C , while actual savings are the difference between C.+ I and C , and therefore always coincide with investmentX« To conclude, savings decisions are simply frustrated as long as they d i f f e r from the predetermined amount of investment. Only when the system has reached the new equilibrium position, do ex ante savings become equal to actual savings. In other words, savings decisions become e f f c t i v e only when the changes in income have made them y i e l d an amount of savings equal to the pre—determined amount of investment. (Pas i n e t t i , op.cit., p. 52—3) Thus we see again how investment determines savings even FIGURE 7.1: Income Expansion With A Lagged M u l t i p l i e r 207 when savings decisions are made before investment decisions, as in the case of the leakage effect of savings discussed above. As opposed to injections, leakages are l i k e l y to have a more immediate e f f e c t . Therefore we s h a l l subtract them as they enter the m u l t i p l i e r . Writing c for the marginal propensity to consume, , t-L as marginal rates of d i r e c t and indirect taxation and m as marginal propensity to consume imported goods, from our c l a s s i f i c a t i o n of leakages as to whether they affect w or p , we can write down their interactions in the following way: n\ and t t a f f e c t C through their operation on w , tj however af f e c t s c by lowering u . Assuming for the moment that t^~Q , we can write the m u l t i p l i e r as follows multipli<.r when t^ >0 ' C w i l l be affected before m and enter into the picture, thus the f u l l leakage e f f c t s from the m u l t i p l i e r can be written as b = 1 . Injections however are also subject to leakages, especially in imported investment goods (T. Wilson, 1968). Then where m* i s the direct leakage in imported investment goods. The importance of this modification i s due to the fact that i f n \ * is of the same magnitude as m in the mu l t i p l i e r AY = r ^ R i - m * ) < J" 208 in other words, i f there i s a s i g n i f i c a n t import leakage in the i n j e c t i o n , then f i n a l multiplied expansion of income could be less than the o r i g i n a l expenditure on investment that gave r i s e to i t (Brownrigg, 1972). Second round investment i s induced by the increased expenditure created by the newly employed, either drawn from the l o c a l area, or immigrants attracted to the area by the increase in incomes there. Let this increase be f\=- J-+G in the area, and A N the t o t a l annual earnings by new employees, then A A = O.AN This method was f i r s t suggested by Archibald (1967) to avoid the situation of having A r W A Y , where "given the formula for the m u l t i p l i e r i t s e l f the denominator may become negative(op.cit., p. 37 see footnote)" (Brownrigg, 1972) However A w i l l also be subject to leakages similar to that of 3", i.e.,m*, therefore We can also divide T into two main elements, the investment expenditure, flowing from injections into the area, Jx , and the second the continuing flow of income a r i s i n g from the p r o j e c t , J 2 . This d i s t i n c t i o n was f i r s t made by Domar, (1948) drawing attention to the fact that not only does investment stimulate income, i t also increases productive capacity, and hence, as Harrod (1939), f i r s t pointed out, f u l l employment requirements grow over time too. (These two contributions form the basis of the Harrod—Domar growth model). Generally speaking, AN w i l l represent a proportion of CT2 varying with the proportion of new labour required by the project created by 3, . Then 209 AN = nj^ and f u l l model may be written as AY = (J1+J i(i+a»0](i~'n*) K 7 . 5 Patterns Of Long Run Urban Growth 7.5a Steady Growth We are now ready to investigate the conditions which must be s a t i s f i e d i f urban growth i s to be steady over time. We have seen how, i f the rate of urban growth i s to keep up with the expansion of productive capacity through investment, the urban labour force must also increase, thus stimulating further rounds of investment. In the Harrod—Domar model, which relates labour supply to productive capacity over time, the rate of investment required to maintain steady growth, i . e . , no overshooting of e f f e c t i v e demand by productivity growth due to investment, is given by ICO = l(0)exp(s/k)t where k - J i , S=JL (•'•&.-§_ =. 5_) * ^ Y K "K K i . e . , investments must expand at a percentage rate of growth so that 210 This i s Domar's contribution. Harrod's i s to enquire what the growth rate of investment should be i f the labour force i s growing at a rate A , and output/man, labour productivity is also growing at a rate "X. The sum of these two rates of growth gives the natural rate of growth 9rv = n + l Then for investment growth to match labour and productivity growth Moreover th i s growth must be exponential. Pasinetti(1962, 1974) and Kaldor (1955, 1966) have used this model to show what the share and rate of p r o f i t in national income must be i f f u l l employment is to be maintained in a c a p i t a l i s t society. The most general version of thi s .model has been given by P a s i n e t t i , (1974). In the Harrod version of the growth model 5, k , andg n are a l l predetermined: in Pasinetti's and Kaldor's development of the model, S i s allowed to vary so that i t becomes As savings of c a p i t a l i s t s are $ = S CP C i S C - saving propensity of c a p i t a l i s t s The share of p r o f i t in national income i s therefore (assume S w = 0 ) y " = s~"5n 7-5.1 When the assumption is made that workers save also Su >0 , and lend their savings to the c a p i t a l i s t at a rate of interest L<TC 21 1 , the rate of p r o f i t s of investors and c a p i t a l i s t s , then the o v e r a l l rate of p r o f i t can be written as follows K = Y^,% ( w h e r e H > 1 > 5 0 t h a t - £ X = SL ) E<£ 7 5 . 3 where 7T = | = - >, 1- > L a n d KT=ir9" (since from 7 . S . 1 , , | = f - = J n ) and )( i s a constant r e f l e c t i n g the influence on the o v e r a l l p r o f i t rate of workers lending to c a p i t a l i s t s and receiving interest on investments in c a p i t a l at a rate i = (U-R , fi-< 1. Where S w « 0 or (ix x JL , 'i = 1 and equation 7 . 5 . 3 develops into equation 7.S .4 (see below, CK«.5) The interest of [equation 7 . 5 . 3 , . writes Pasinetti] is that i t shows that t comes to reinforce s<_ . A rate of interest lower than the rate of p r o f i t has the same effect as a higher propensity to save of the c a p i t a l i s t s , as i t redistributes income in favour of the workers This is exactly what we would expect from Keynes' account of investment behaviour: the larger the gap between the rate of interest to the rate of p r o f i t , the greater the volume of investment. Equation 7 . 5 . 3 can also be generalised to include landlords as a separate c l a s s . Pasinetti ( 1 9 7 7 ) writes: [By] considering landlords and c a p i t a l i s t s to—gether and adding t o t a l rent to c a p i t a l i s t s ' p r o f i t s in the equations on pp. 1 1 0 - 1 1 of my book, one 212 ends up with the equation where r is the long run rate of p r o f i t , gn is the "natural rate of growth", St is the l a n d l o r d — c a p i t a l i s t s propensity to save, and S is the long run r a t i o of t o t a l rent to c a p i t a l i s t s ' p r o f i t s . This r a t i o simply comes here to reinforce sc. The only condition that must be s a t i s f i e d i s that t o t a l rent i t s e l f enters the savings relations in a constant proportion, otherwise a steady state path would not e x i s t , and we would be outside the scope of this sort of analysis. Thus a f u l l y s p e c i f i e d version of equation 7.6.3, with both landlords, c a p i t a l i s t s and workers who lend their savings to c a p i t a l i s t s in return for a rate of interest I, would be as follows Kaldor concludes from this r e s u l t , and from his views on increasing returns to scale that in the long run therefore the only constraint on the growth of an econom^.: in long run equilibrium i s population growth. Pettenati (1975) however argues that such an interpretation i s inconsistent with Keynes' theory of interest and money. In a (more or less) fixed c o — e f f i c i e n t s world, f u l l employment can mean either f u l l capacity c a p i t a l u t i l i s a t i o n , or f u l l labour employment. Kaldor chooses labour as the constraint so that "Y ^ (where ^ L , y K are the levels of income corresponding to f u l l employment and f u l l capacity u t i l i s a t i o n , r e s p ectively). However, Pettenati shows that i f Y , i . e . , labour a v a i l a b i l i t y i s the constraint, the interest rate must be determined by the real sector, the l e v e l of prices by the monetary sector, and the 213 d i s t r i b u t i o n of income by the degree of monopoly prevalent in the economy. independently of the level of output. As we have seen, Keynesian theory demands that the interest rate be a monetary phenomenon, the l e v e l of prices a function of the rate of investment, and the real wage a function of the investment determined l e v e l of output, therefore Kaldor's interpretation of the long run constraints of his theory is not consistent with a Keynesian short period analysis. However the constraint that — K — L Y ' ^ s o n e that is consistent with a post-Keynesian theory of d i s t r i b u t i o n , and accords with Keynes' argument that c a p i t a l commands a price, not because i t i s productive, but because i t is scarce (Keynes, 1936, ch. 16). Keynes also demonstrates why in a c a p i t a l i s t society, c a p i t a l must be kept scarce, because i t must y i e l d a rate of interest at least equal to the rate of in t e r e s t . Thus we may say that an i n s t i t u t i o n a l e f f e c t of c a p i t a l i s t production is to keep c a p i t a l scarce. Although the rate of p r o f i t in long run equilibrium depends on g n , l e t i t be remembered that Cj^ is the sum both of the rate of growth of the labour force, and of the rate of growth of labour productivity. Labour productivity, however depends by and large on the technical progress, embodied of course in new c a p i t a l . C n^ therefore depends as much on the rate of growth of new c a p i t a l , as on the rate of growth of the labour force (that i s , i f c a p i t a l a v a i l a b i l i t y i s the constraint, and labour depends on technical progress embodied in the new c a p i t a l ) . Kaldor's discussions begin with the "simple" case in which X c Q 214 and this may well have been a misleading beginning. For i f we argue with Boserup (1965) and Wilkinson (1973) that, ultimately population pressure i s the catalyst of technical change, then rv>0 =S> A > 0 also. Naturally the s o c i a l conditions under which technological change proceeds cannot be ignored. Eichner (1976) writes of The tendency in economic model building to treat i t [technological change) as being e n t i r e l y exogeneous — a windfall from s o c i a l processes outside the economic system. To the extent that technological progress depends ultimately... on the growth of s c i e n t i f i c knowledge, there is perhaps some j u s t i f i c a t i o n for this approach — even though the a b i l i t y of a society to support some of i t s members while they pursue their idle c u r i o s i t y i s not e n t i r e l y independent of the margin above subsistence which that society is capable of generating nor of the pattern in which the surplus i t s e l f i s generated (p. 231). There i s a further problem with assuming a steady rate of technical progress. The idea that one can project such a rate into the future is to f a l l into the h i s t o r i c i s t f a l l a c y (Howard and King, 1975). In fact one of the effects of the process of technological change i s to increase uncertainty about the future, since i t i s impossible to know what the future d i r e c t i o n of that progress w i l l be. The constraint on investment i s paradoxically that which is i t s impetus. There i s an impetus to invest in new techniques to gain a competitive edge over one's r i v a l s : however there i s the counterbalancing fear that technical change instigated by a competitor w i l l render the presently new technology obsolete. Heavy investment in a now outdated style w i l l result in an i n a b i l i t y to turn those assets 215 back into cash at the time that payment for the investment f a l l s due. In an economy,: where firms can to a certain extent control the pace of their investment therefore we may say something about the constraints on investment, and on the shape of competition among such firms. Competition w i l l tend to be o l i g o p o l i s t i c , taking place not through price responses, but through investment. Eichner i l l u s t r a t e s this point by reference to the a i r l i n e industry, which faced with the development of the jet a i r c r a f t would have preferred to wait u n t i l i t s existing stock of propellor driven planes was f u l l y depreciated before converting i t s f l e e t s to a l l j e t s . The in t e r f i r m competition, however, meant that once one a i r l i n e had purchased the jets, the others had to follow suit even though i t led to considerable di f f i c u l t y . Capozza and Van Order's (1978) c l a s s i f i c a t i o n of competitors' reactions to price changes thus probably corresponds to the varying sizes of competitors, o l i g o p o l i s t i c competition producing less conjectural variation than p o l y p o l i s t i c competition. The forgoing discussion of the role of technological progress in shaping the growth of the economic system has been couched completely in aggregate, national macro—economic terms. Those seeking to use t h i s approach in an urban context have t r i e d to deal with i t by treating the urban economy as i f i t were a small country with open borders thus, while i t i s clear that from the viewpoint of the economic as a whole, technical progress must be regarded as being endogeneously created, i t i s 216 argued that in the case of the urban economy, that technical progress i s more l i k e l y to be exogeneously than endogoneously determined. Therefore i t i s more permissible to impose technical progress as something occurring from without, even i f i t means that the parameters of the system have to be recalibrated in every period as the course of technical progress proceeds. Furthermore, to return to our discussion of CJ^ being composed of H.+ A , i t i s clear that rv cannot be imposed as given in an urban economy. At any time, fl w i l l be the sum of the natural b i r t h rate of the urban c i t i z e n s , plus any net in—migration as a result of changing economic conditions in the urban area, which are l i k e l y to be of a s i g n i f i c a n t order of magnitude. Given the openness of the urban economy, this leaves technical progress therefore, i f only by default, the only independent variable defining ^/^, since n w i l l accommodate i t s e l f to /^|^ . In long run equilibrium, on these assumptions then, the rate of growth of the urban economy i s given by eqn. 7.6.3 as is that of the national economy. However the rate of growth of the urban population w i l l be given by ^ C/K ' and I w i l l be constant in long run equilibrium, and since ^c/K C i s constant, "\ w i l l be constant also, because technical progress, and thus labour productivity can a l t e r only as the pace of investment a l t e r s . Thus we have one equation and one unknown. Furthermore the Sc term w i l l r e f l e c t the c a p i t a l i s t s ' interest in keeping c a p i t a l scarce enough to y i e l d 217 a p r o f i t greater than the ru l i n g rate of interest. The next issue to be resolved i s the rate of growth of the urban area in long run equilibrium. The simplest way of deriving such a growth rate is to say that the addition of each extra c i t i z e n requires a certain average amount of land, say <X, then t o t a l land area A , w i l l equal, at time t However i t is l i k e l y that c< also w i l l grow over time with technical progress and r i s i n g incomes. New houses w i l l be larger also. Therefore i t i s best to make c< a growing function over time. One way i s to incorporate an exponential growth function, then ft It) = et\ M U and the rate of growth of the land area where i s the derivative of (HO with respect to time. Let us repeat that these are the conditions which w i l l hold i f long run equilibrium i s attained. They are not l i m i t i n g factors to which economic factors necessarily tend. They are what would occur i f investment and productivity growth never overshoot e f f e c t i v e demand. The following section gives an account of what, on these assumptions, happens when this does occur. 7.5b C y c l i c a l Growth If the response of investors to r i s i n g incomes i s to 218 increase their rate of investment, then there is always the p o s s i b i l i t y that they may over—estimate the extent to which incomes w i l l r i s e , and thus be led into providing a greater productive capacity than i s in fact suited to the actual l e v e l of income at the time that the new investment comes into use. This s i t u a t i o n in fact recreates the problem of i n s u f f i c i e n t e f f e c t i v e demand, but at a higher l e v e l of output and income than before. M u l t i p l i e r — a c c e l e r a t o r models of the type discussed by Allen(1956), Yamane(1968), Pasinetti (1974) and Richardson (1969). The investment accelerator states that investment i s a function of expected future income, Yf However, since the only key to future income is that of past changes in income, change in income from the preceding period i s used as a proxy variable for expectations of future changes in income: where 0^ i s a behavioural parameter, the desired capital—output r a t i o , i . e . , the desired r a t i o between the existing c a p i t a l stock and production per unit of time (Pasinetti, 1974). Investment in any period also equals the change in c a p i t a l stock from the preceding period I = fiw*) therefore change in investment is given by 2 1 9 which gives (and can be interpreted as the desired stock of c a p i t a l . Investors w i l l seek to carry out investment (or disinvestment) whenever their existing c a p i t a l stocks d i f f e r from the amounts they desire to have. However they may spread their investment over a number of time periods. Let the number of periods equal V(J, then It = P^Vi' K t J 0<(H1 or Investment i s a linear positive function of income and a negative function of the existing c a p i t a l stock. Before we can combine th i s function with the m u l t i p l i e r , we need to make some modifications to the Keynesian system of macro—economic accounting in order to take into account the leakages and injections previously discussed with respect to the urban m u l t i p l i e r . F i r s t i t i t i s clear that the i n i t i a l i n j e c t i o n , 3^  > i s determined by tfj. = T ^ l o E^i.«b.i C = C(_l ~ ^  C i - m ~ ^ l ) / W - marginal propensity to import c a p i t a l goods 220 so that 3^ = t J t _ 0 =vJ 0 in period analysis ( i . e . , the i n i t i a l i n j e c t i o n , which here has a time subscript of zero, i s given by Brownr igg' s \JX ) However, in subsequent time periods as might be expected at f i r s t glance, because CT^ consists of both public and private investment. Though private investment w i l l respond according to the accelerator mechanism, we cannot assume that public investment w i l l also. Thus 3"t = jTS* Vi~ Kfc-i + I^T* t^ + Xt Ec^ 7.Sb.3 Remember that St included only I + Q, but that in an urban economy imports and exports w i l l be very important. Bearing t h i s in mind, we may now write out a set of equations defining the urban economy, using the superscript I to refer to the urban area, and j to refer to the rest of the economy ( i - j ^ ^ j ) , a t i l ~ over the l e t t e r ' t ' to refer to marginal tax rates, and a pla i n ' t ' for time. We may now define regional income, injections, leakages, consumption and investment. C t ' = C< + c Y t l . i w T S b 5 221 Investment in I is the sum of domestic and outside investment decisions actually implemented in period t l\ = I f ttf 7.5b-6 I ° t = m t . l j +It E * a 7 . 5 b . 7 ( I t i s the homogeneous term in th i s linear approximation and incorporates technical progress over time) Imports and exports are functions of the levels of of demand for j ' s and I's respective consumer goods, governed by the le v e l of income in I and j respectively, plus that part of investment which requires imports of c a p i t a l goods from j or I (= exports from I o r j ) plus that part of government expenditure in the region which i s used to import outside produced goods, n\j^  i s the weighted average of propensity to import both c a p i t a l and consumer goods (generally fllyL < W ^  ). M i l' = m j L Y t L : 1 + m*siILbm + w i l C J t £<^7 .5b .$ X ' j = m i j ^ i - i + "Is1'*- E ^ i s b J We use fn-- rather than some some expression for the marginal <i propensity to export, since exports do not depend on one's p r o c l i v i t y for exporting but on the state of demand elsewhere. 10 - r 1 - i / i if1 E<lQ 7.5b.II T> - 5 y _ V J + - & - + - i r l t E V 7-5b-i3 222 This presentation i s more f l e x i b l e than that of Richardson's (1971) account of an inter—regional business cycle. In an open national economy there is no reason a p r i o r i why import requirements from j to t should be equalled by exports from I to J , i . e . , that 1 - m j i = mLj fyfl- 7.5b. lit However Richardson makes t h i s assumption when he writes his constraint as % l - l I S 1 which would imply 1 - 2 m - C^-l "because m ; . Was no mamma ) which reduces to 7.Sb.14 in the case of tl = 2. Such a "balance of trade" constraint makes l i t t l e sense in the context of a national economy with a public sector, i . e . , in the context of a government sector with taxing and spending powers. Richardson does not include taxation and writes merely G l t = G l i . e . , a constant. Now i f vti. - Q C Y ; ) l e t us say as a f i r s t approximation, taking leakages into account Ci '« (5 i + g Y t i . 1 ) ( i - « j i ) e^7.5b . is (where Gj, is that part of national expenditure spent in i which is associated with public or c o l l e c t i v e goods, e.g., defence, or information) and then government expenditures can be looked to make up the 223 difference between exports on the one hand, and imports on the other. However i f we allow for discretionary spending by governments, then they may more than make up the difference, just make up the difference, or allow the area to suffer a real loss in income, possibly in the hopes of encouraging expansion elsewhere. Then X U + GfJL = l L ( M i l + T 7.5b.17 0 r 1 1 ' — u 0 < $ 1 , depeniina on the state ei nJl4T,'J governtrttnt policy We thus have 10 variables Yl Y\ K\ KJ, X'j, MJ,L GJl, TU, I', I J , and 10 equations 4, 8, 9, 10, 11, 12, 13,-35, 16 and 17. The system i s thus determinate. Substituting 13 into 9, and 10 into 8, putting the resu l t i n g expressions and 5 into 16 and then 5, 9, 8, 15 and 16 into 4 gives * It w i l l be noticed that t h i s equation contains terms in Y and K J which are exogeneous to the system. These are a l l in however. Re— arranging and substituting 17 into 4 and 18 gives 224 C jl also drops out of the equation, but helps govern U as discussed above. Writing Mb = [ l ( i + Cu--i)fcdCL)+a^ ) _^mIy^  I j t ^i5b.ao we can re—arrange 19 to give Then writing we can re—arrange 21 for rH-1 to give = J - (Pc • S*,) - |; Yj + J - Vt £ ^ 7 . s b . t t substituting 11 into 19 gives Y; = + + K - + ^ - ^ A V ^ K - K U ) ] + 'oi-i )Cr l l (ai i -cY;. 1)+^(KJ-KU) ^ , . 5 ^ re—arranging for Kfc and substituting 22 for ^t-i 9 i v e s + JL Vt- (l-Ot-lYfj^Cj fcy5 TSb.U where ^ = 1+ (CL-IX™ * . . + fc») (as a.-, (i. drop out of the system, we drop the subscripts for ot. and p^also) 225 Combining the two constant terms in 24 yiel d s writing and ^ we can write 23 as Substituting 25 into 24 and 24 into 23 for Kf, and then s h i f t i n g 24 back one period and repeating the procedure for gives YLT = N(W-|)Y; T {_ N(tf-R ) + C + R + fl£c+fi+^K(M«3}Y* Writing X = l'^^""-^), we can re—arrange 26 giving f i n a l l y i . e . , a non—homogeneous, l i n e a r , second order difference equation with constant c o — e f f i c i e n t s . The pattern of growth that t h i s model w i l l exhibit w i l l vary from exponential growth (or decline), to explosive (or damped) o s c i l l a t i o n s , both types of movement taking place with 226 respect to- a trend path given by the technical progress function jVVfc + Ci* ( i 'a ) t ( J L ) C f Which type of growth occurs w i l l depend on the values taken by the parameters of the system. Some of the model's c h a r a c t e r i s t i c s are outlined below. As t h i s i s a non—homogeneous model, i t has both a par t i c u l a r and a general solution for the value of Y over time. The p a r t i c u l a r solution i s where — __ _ _ r. = z.[H-R(w~|) + c + W(c+ H + ^ K ( P c + S o c ) ] ] and The general solution i s then of the form where and are arb i t r a r y constants given by the i n i t i a l conditions of the system. They may be evaluated by obtaining a form for of the type ' ^ in which case 227 X 5C-1-X, and o __ Ko - X i Yp (see P a s i n e t t i , 1974 for d e t a i l s ) . If !72">/(-f^  , \ X i x w i l 1 be real numbers, and growth or decline w i l l be exponential and steady. If I ^ ^ k Q . , X 1 ( i w i l l be complex, conjugate numbers, and growth w i l l be fluctuating. The conditions under which ^ A-Q are not investigated here. Owing to the large number of parameters involved, simulation rather than analysis should be used to determine these. However, a couple of observations may be made about the ranges of values which these parameters are l i k e l y to have. F i r s t , in order that growth may be s e l f sustaining, the marginal propensity to invest must be greater than the marginal propensity to save, i . e . , oC > 1- C Second, in order to avoid the p o s s i b i l i t y of rectangular o s c i l l a t i o n s , must be p o s i t i v e . This means that r;< o 4=> I/OI-AC)! > l c l Again, the values of the parameters required for t h i s condition to hold are not investigated here. One thing that may be said however, i s that the value of i s l i k e l y to be c r u c i a l in deciding the path of income growth over time. Recall that i s the timing of investment, or c a p i t a l stock adjustment parameter. It is the product of the marginal response of entrepreneurs to changes in income, and the desired capital—output r a t i o . If the 228 uncertainties surrounding future le v e l s of demand and the course of technical progress affect each of these factors separately, they w i l l a f f e c t their product in due proportion. As Keynes (1936) said, i t i s normal to regard that variable which i s prone to the most and violent fluctuations as the causa causans of the system. However, as ^ enters into the system of equations both d i r e c t l y and inversely (*t\ — p- |Q(Qft-ft)^  ' * t s effects are not simple. In the situation where the values of the parameters are such as to give fluctuating growth, a graph of Yt with time might such as given by f i g . 7.2. As can be seen, cumulative growth is accounted for by the technical progress function, away from this l i n e . The major problem with t h i s approach stems from the c h a r a c t e r i s t i c s of the investment accelerator. This should not be surprising, since as Pasinetti points out, the assumptions underlying the use of the income m u l t i p l i e r and the investment accelerator are at two -quite d i f f e r e n t l e v e l s of abstraction, the one concerning the accelerator r e l a t i o n being the more remote of the two. The d i f f i c u l t y here is that one cannot hope to take th i s into account by any mathematical formulation. When [the two relations] are written side by side and joint elaborations c a r r i e d out, these can make no d i s t i n c t i o n between the parameters coming in from r e l a t i o n and those coming in from the other. A l l parameters are treated in the same way, though the assumptions behind them have a quite d i f f e r e n t r e l i a b i l i t y . ( P a s i n e t t i , 1974, p. 50) The problem is. that the c h a r a c t e r i s t i c s of the investment Steady exponential growth w i l l also occur accelerator have to be treated as being constant over time. 229 ex, 230 Despite the argument that technical change i s l i k e l y to be exogeneously given in an urban context, the c r u c i a l variable, and one which cannot be modelled, i s the entrepreneur's assessment of the course of technical progress and i t s concomitant e f f e c t s . This i s what the parameter |i attempts to capture. Since the model s p e c i f i e s the future course of technical progress, this c h a r a c t e r i s t i c quality of cannot be maintained. If growth continues to o s c i l l a t e round the technical progress function, the implication i s that entrepreneurs never learn from past experience. If they do learn and adjust their investment behaviour to meet the requirements of the technical progress function, their investment behaviour w i l l become a constant and their expectations r e a l i s e d for a l l time. Thus the model would return us to the world of neo-classical economics where i f time i s brought in, i t i s as a l o g i c a l construct only (Robinson, 1978), where movement backwards in time i s as easy as movement forwards, and where movement forward i s given as completely as completely as the history of the economic system up to the present moment. To keep the technical progress function in this model is to f a l l into the h i s t o r i c i s t f a l l a c y , a f a l l a c y summarised by Howard and King (1975) H i s t o r i c i s t s f a i l to see that any h i s t o r i c a l trend must depend on certain h i s t o r i c a l conditions which may not persist into the future. We cannot know that they w i l l persist and therefore cannot c l a s s i f y anything as inevitable. If we accept that knowledge af f e c t s the course of history, then in order to predict the future with certainty, we must know the future development of knowledge. This Popper observes i s l o g i c a l l y impossible, for then such knowledge would 231 not then l i e in the future. As Pasinetti (1974), Kaldor (1972), Shackle (1972) and Radner (1969) a l l argue, where there are economies of scale, growth in output w i l l be attended by changes in the organisation of production, changes which under the heading of technical change, the future development of which is fundamentally uncertain. By uncertain knowledge l e t me explain, I do not mean merely to distinguish what i s known for certain from what i s merely probable. The game of roulette i s not subject in t h i s sense to uncertainty; nor is the prospect of a victory bond being drawn. Or again, the expectation of l i f e i s only s l i g h t l y uncertain. Even the weather is only moderately uncertain. The sense in which I am using the term i s that in which the prospect of a European war i s uncertain, or the price of copper and the rate of interest twenty years hence, or the obolescence of a new invention or the position of private wealth holders in the s o c i a l system in 1970. About these matters there i s no s c i e n t i f i c basis on which to form any calculable p r o b a b i l i t y whatever. We simply do not know. (Keynes, 1937) But as Pasinetti argues there are two aspects to technical progress. One i s the reorganisation of production concomitant with increases in the volume of output. The other is that the "meaning for the community as a whole of a l l improvements amounts to a continuous growth of real per capita income." Pasinetti writes The second of these aspects has been completely neglected by a l l macro—economic models, and i t comes to assume a p a r t i c u l a r relevance in our analysis. It i s , indeed, an inherent c h a r a c t e r i s t i c of economic behaviour, already discovered in the last century by E. Engel, and recently confirmed and i n s i s t e d on by a l l economists who have dealt with quantative evaluations of demand, that when real per capita 232 income increases, the tendency of consumers i s n o t to d i s t r i b u t e the new income proportionally among the commodities previously bought, but on the contrary, to dir e c t the extra demand to—wards new goods... As a consequence of thi s behaviour, the r e l a t i v e composition of the purchases of consumers varies in time. For the economic system as a whole, the effects are not only that the employment structure changes... but that the r e l a t i v e composition of the national product, in real terms, continuously changes, and thi s as a dire c t consequence of technical progress. [we] have come here to an important l i n e in [the] the argument. The nonproportion growth of demand faces entrepreneurs with a big problem, a problem which would not exist in the case of a proportional growth of the whole system. The problem consists in finding out those productive branches (always d i f f e r e n t as income increases) which correspond to the next consumers' preferences in the process of demand expansion caused by an increasing per capita income. The problem also e n t a i l s the adaptation process of consumers to higher p o s s i b i l i t i e s of consumption, and is complicated by the choice that a growing system i s continuously facing, between that part" of productivity gains which i t prefers devoting to higher production, and that, which i t prefers devoting to l e i s u r e . Evidently, in the attempts to fi n d out the solution to th i s complex problem the entrepreneurs may make mistakes, and in some periods their fear of being mistaken may be higher in some periods than in others, resu l t i n g in hesitations and postponements of investment projects. At a macro—economic l e v e l , the meaning of these mistakes or simply hesitations comes down to a change in the aggregate behavioural parameters of the system. This has a decisive effect on the dynamic movements of ef f e c t i v e demand. (Pasine t t i , 1974 pp. 73-74) Thus i t i s the circumstances surrounding the investment decision, i . e . , (3, and not the some inherent c h a r a c t e r i s t i c of the multiplier—accelerator model which explains the outstanding feature of our actual experience—namely that we o s c i l l a t e , avoiding the gravest extremes of fluctuation in employment and prices in both directions, round an intermediate position appreciably below f u l l employment and 233 appreciably above the minimum employment a decline below which would endanger l i f e . (Keynes, 1936, p. 254) Pasinetti's remarks concerning the effects of technical progress on demand also raise an issue for the neo—Ricardians. Although we have said that the Sraffa system requires constant returns to scale, Sraffa himself argues that i t does not, and that Keynes t o l d him in 1927 that he must make t h i s quite clear. However, as Robinson argues, i f constant returns to scale are not assumed, then i t must be assumed that the composition of output (and the demand for goods) does not change with changes in the scale of product in the Sraffa system. However, the quotation from P a s i n e t t i , and also our previous discussion of Engel curves in re l a t i o n to u t i l i t y theory, makes i t clear that the assumption of of no change in the composition of demand for output is merely the other side of the coin from assuming constant returns to scale. The multiplier—accelerator model therefore has only limited usefulness, even at the macro—economic l e v e l , to say nothing of i t s application to the urban s i t u a t i o n . Even i f we were to drop the technical progress function (so that eqn. 7.6b.27 becomes homogeneous, and equal to zero), the model would need to be recalibrated in every period to take into account the changes in knowledge and attitudes to—ward the future engendered by the events of the previous period. However such a model does have a similar pedagogic use similar to that of the Harrod—Domar model of steady growth presented in the previous section. In this case i t s usefulness 234 i s of the "what would happen i f " variety. Depending on the values taken by , there may be growth or there may be fluctuations, but as can only take on one value at a time, the occurence of both growth and fluctuations simultaneously cannot be accounted by th i s model. Since however the parameter (whose fluctuations have been i d e n t i f i e d as being the main reason for the system switching from one state to the other) i s governed by expectations about the e s s e n t i a l l y unknowable future, the the interaction between the m u l t i p l i e r and the accelerator c 1 n account for the fluctuating growth t y p i c a l of a c a p i t a l i s t economic (by assuming that ^ f a l l s f i r s t in a range that produces the one state, and in a range that produces the other) so long as the conclusions drawn.are not pushed too far. To quote from Pasinetti once more A l l this amounts to i s that the results must be interpreted with care and judgement. If the analysis of the interaction of the m u l t i p l i e r and the accelerator has taken the theory of e f f e c t i v e demand to i t s peak, i t has also c l e a r l y strained i t to the l i m i t s of i t s p o s s i b i l i t i e s . For as soon as economic investigation is car r i e d over from the Keynesian short run to movements through time, and the c a p i t a l stock can no longer be taken as given, but i s i t s e l f changing, then another part of the picture i s bound to become relevant — the evolution through time of the physical p o s s i b i l i t i e s of production. It is the f a i l u r e to appreciate the l i m i t a t i o n s inherent in thi s type of economic modelling which r e a l l y distinguishes the neo—Keynesian position from that of the post-Keynesian. However, as we s h a l l show below., applying Keynesian theory to the urban situ a t i o n i s not best approached through viewing the c i t y as a 235 small country with open borders, rather i t is necessary to show how Keynesian theory can be disaggregated^ not merely scaled down, as with the two "post-Keynesian" models presented here. However as a f i n a l preliminary to t h i s issue, we now turn to the question of autonomous and induced investment. 7.6 Autonomous And Induced Investment Hicks (1950) in his "Contribution to the @theory of the Trade Cycle" suggests that we can make a further d i v i s i o n of investment, into induced investment and autonomous investment, with induced investment dependent on income. From th i s we may derive a 'super—multiplier' as follows C + I investment, X d = autonomous investment, S) = marginal propensity to invest. Then and proceeding as in the case of the simple m u l t i p l i e r 1 236 and 1 _ L * l - ^ ' + ' v ' ) ~ t n e s u p e r - m u l t i p l i e r Two o b j e c t i o n s may be made t o t h i s p r o c e d u r e . F i r s t t o make p a r t of investment dependent on c u r r e n t income i s t o make investment dependent a t l e a s t i n p a r t on s a v i n g s out of c u r r e n t income, s i n c e Xvj can o n l y be f i n a n c e d out of t h a t p a r t of c u r r e n t income which i s saved. However, i f any p a r t of investment i s made dependent on s a v i n g , a l l investment i s dependent on s a v i n g , s i n c e s a v i n g has now been p l a c e d p r i o r t o in v e s t m e n t , from which i t f o l l o w s t h a t S - 5 > X . S e c o n d l y , however when we come t o ask what i n c u r r e n t income would induce such i n v e s t m e n t , the answer can o n l y be the requirement of m a i n t a i n i n g c u r r e n t consumption, by maintenance of investment i n machinery t o keep the consumption s e c t o r i n a s e l f — r e p r o d u c i n g s t a t e . Thus we can e i t h e r w r i t e Y = C + i = K Y M = 4 ^ 1 or we can w r i t e and as Y = Y, v(Y) = KS), then M* = ?' and t h e r e f o r e c-- Is Induced investment must e q u a l c u r r e n t consumption i n the i n s t a n t a n e o u s m u l t i p l i e r . Induced investment i n i n l a t e r rounds i s merely a case of the lag g e d m u l t i p l i e r . In the p r e v i o u s s e c t i o n , we i d e n t i f i e d the r e l a t i o n oC>\~C as b e i n g a n e c e s s a r y c o n d i t i o n f o r growth i n the system. T h i s i s e q u i v a l e n t t o s a y i n g t h a t the m a r g i n a l p r o p e n s i t y t o i n v e s t 237 must be greater than the marginal propensity to save. Here we appear to be asserting a further condition, namely that oC. - \) - C However in Hicks' system, as we have treated i t , the marginal propensity to invest enters the system not through the accelerator, but by way of the m u l t i p l i e r . When the investment decision is based on estimates of future income, rather than current income, then the aggregate propensity to invest need not, in fact w i l l not be the same as the aggregate propensity to save, so we need not r e s t r i c t oC to equality with C. Either way however, the result s t i l l holds, that investment decisions "drag" savings decisions along aft e r them in a manner consistent with the logic of Keynes's system, in a similar manner to that described in the operation of the lagged m u l t i p l i e r . We now turn to an examination of the problems posed for economic theory by the existence of s p a t i a l factors. These turn out to pose many of the same problems occasioned by the need to to treat time in economic theory in a n o n t r i v i a l way, and confirm the a b i l i t y of post-Keynesian theory to handle these better than the alternatives reviewed here. 238 8 POST-KEYNESIAN APPROACHES TO URBAN ANALYSIS 8.1 Space And Economic Theory A space economy is characterised by the presence of economies of scale, plus the existence of transport costs (Capozza and Van Order). Of transport costs (Capozza and Van Order, 1978). As they put the problem: The rationale for the study of space i s f i r s t that c a p i t a l i s lumpy. If i t were perfectly d i v i s i b l e , with no economies of scale, there would be no location problem. Secondly, i f transport costs tend to—ward zero, space becomes an irrelevancy. With the existence of positive transport costs, firms exhibit the downward sloping demand curves c h a r a c t e r i s t i c of monopolistic competition. (Chamberlin in fact uses a s p a t i a l analogy in the second edition (1956) of his work on monopolistic competition) [such a] combination of increasing returns and competition i s a very prominent of decentralised economic systems — a circumstance in which each producer faces a limited market as regards sales, and 239 yet a highly competitive market as regards prices. (Kaldor, 1972) Capozza and Van Order i l l u s t r a t e some of the counterintuitive ways in which this competition may manifest i t s e l f . For example, since sales are conducted over market areas in which the price e l a s t i c i t y of demand w i l l increase with distance, exit of firms may be associated with r i s e in prices, depending on the assumptions as to competitors' reactions to price changes. This is contrary to the predictions of nonspatial price theory, which has exit of firms associated with a f a l l in prices. However as Kaldor argues, the existence of economies of scale has implications not only for neo-classical p r i c i n g theory, but other aspects of neo-classical economics also. Kaldor argues that where economies of scale ex i s t , real income i s limited solely by e f f e c t i v e demand and constraints on growth, and not at a l l by constraints on resources. In such a case, when every change in the use of resources — every re—organisat ion of economic a c t i v i t y creates opportunity for a further change, which wou Id not have existed otherwise, the notion of an optimum al l o c a t i o n of resources becomes an meaningless and contradictory one: the pattern in use of resources at any one time can be no more than a link in the chain of an unending sequence, and the very d i s t i n c t i o n between resource creation and resource a l l o c a t i o n , v i t a l to marginalist equilibrium economics loses i t s v a l i d i t y . (Kaldor op.cit. ) Thus the influence of space on the economy, which renders a l l competition inevitably monopolistic, yields results which are inimical to neo-classical theory. They are also inimical to 240 neo-Ricardian theory however, which shares with neo-classical theory the same assumptions as to competition, returns to scale and the role of money in a c a p i t a l i s t economy. None of these issues however are problems for post-Keynesian theory, to quote from Kaldor once more. The existence of increasing returns requires above a l l a monetary and a banking system that enables c a p i t a l investment to increase in response to inducements so as to generate the savings required to finance investments out of additions toproduction and incomes. This is the real significance of the invention of paper money and of credit creation through the banking system. As we have seen, money i s an essential feature of the Keynesian account of the economic system. We are now ready to examine how the orthodox, micro—economic account of the price system explains, or rather, f a i l s to explain the operation of the housing market. 241 8.2 The Orthodox Theory Of Housing Supply And Demand The orthodox theory approaches the issue of housing supply and demand from a s t r i c t l y micro—economic (and thus neo-classical) standpoint. There appears to be an unquestioned(ing?) consensus of professional opinion on t h i s subject, regardless of any other differences of opinion on economic issues that those treating t h i s subject may have. Thus the Fraser Institute has been able to c o l l e c t statements from such a diverse group of economicists as von Hayek and Friedmann on the right, to Myrdal and Lindbeck on the l e f t , a l l opposing rent control on the basis of micro—analyses of housing market behaviour (Fraser I n s t i t u t e , 1975, 1981) even those who see a need for rent control on 'social'grounds are forced to admit the adverse 'economic' consequences of such controls, with the result that, l i k e Keynes himself in 1926, their 'sympathy and their judgements are l i a b l e to be on d i f f e r e n t sides, which is a painful and paralysing state of mind' (Keynes, 1926, quoted in Minsky, 1975) Since the issue of rent control i s so c l o s e l y t i e d up with the orthodox method of analysing housing market behaviour, we s h a l l use t h i s issue as a peg on which to hang a discussion of the orthodox theory. The model used by opponents of rent control in their discussions of the subject i s along the l i n e s of that shown in f i g . 8.1a. It i s assumed that supply of housing in the short period is fixed at , y i e l d i n g a short run supply curve of CM CN 14. i 14. i i Pe 8, G. 11 » 1 / 7 / y -/v\-a — * new construction XLG.URE 8.1 :...i. i_lhe_Analysis of Rent Con t r o l According to Micro-Economic P r i c e Theory i i . The R e l a t i o n of New Housing Construction to Rent Control 243 Q ^ i S g . The equilibrium price p e i s given where the demand curve intersects the short run supply curve. If , however, rent control maintains the price of housing at r ^ instead of allowing i t to r i s e to the equilibrium price of , the rate of supply of new housing w i l l be reduced from to In equilibrium, the point £ i s also intersected by the long run supply curve yD^S |_ which, since e l a s t i c i t y of supply i s greater the longer the period considered, has a lesser slope than that of the short period supply curve. The long run effect of maintaining the price at JD^ i s then to reduce the supply to Q^, whereas demand at the price w i l l have increased to Q\ s . Thus the effect of rent control i s to create a gap between the amount producers are w i l l i n g to supply, and the amount purchasers wish to consume. Therefore i t i s argued that rent controls, usually imposed to prevent landlords from u n f a i r l y exploiting a shortage, in fact perpetuate such shortages. This in turn encourages immobility of housholds (and therefore also labour), since so few vacancies w i l l come onto the market. This prevents the market place from performing i t s function of responding to consumer desires via the p r o f i t motive and the price signaling mechanism. Supply of services from the existing stock is further reduced since '"--..landlords wish ing to maintain their rate of p r o f i t , w i l l reduce their expenditures on maintenance, in addition to the erosion of production incentives due to the reduced rate of p r o f i t on new stock. If the price signals 244 generated by the market are also supposed to be responsible for the a l l o c a t i o n of factors of production, then resource a l l o c a t i o n and also the pattern of labour use w i l l also be distorted. (Pennance, 1975, in Fraser Institute, 1975) The solution i s then to free housing from rent controls since the the consequent rise in rents w i l l have w i l l have two eff e c t s . F i r s t , t h i s w i l l choke off excess demand. Second, i t w i l l make investment in . residential construction more favourable. Thus although high rents charged on those already in rental accommodation means that those renters w i l l in effect subsidise new construction for others' benefit, the increase in supply thus engendered, w i l l in the long run be of benefit to a l l (Walker in Fraser I n s t i t u t e , 1981) The question to be answered however is whether this damning li t a n y of complaints against rent control is in fact the real reason for a persistent shortage and even actual decline in the amount of rental accommodation on offe r , or whether these allegations are a consequence of the method of analysis employed. The assumptions necessary in order to apply t h i s type of analysis are given by 01sen(l969). Let us assume that the following conditions are s a t i s f i e d in markets for housing service: (1) both buyers and s e l l e r s of housing service are numerous, (2) the sales or purchases of each individual unit are small in rel a t i o n to the aggregate volume of transactions, (3) neither buyers or s e l l e r s collude, (4) entry into and exit from the market are free for both . producers and consumers, (5) both producers and consumers possess perfect knowledge about the 245 p r e v a i l i n g price and current bids, and they take every opportunity to increase production and u t i l i t y respectively, (6) no a r t i f i c i a l r e s t r i c t i o n s are placed on demands for, supplies fo. And prices of housing service and the resources used to produce housing service, and (7) housing service i s a homogeneous commodity... Most scholars would probably find (7) to be the least plausible assumption... Many presume that a very heterogeneous good i s traded in the housing market... Therefore, we w i l l focus our attention on this c r u c i a l simplifying assumption. In order to view the housing market as one in which a homogeneous commodity can be bought and sold, an unobservable theore t i c a l entity c a l l e d housing service i s introduced. Each dwelling (or housing) unit is presumed to y i e l d some quantity of t h i s good during each time period. It i s assumed to be the only thing in a dwelling to which consumers attach value. Consequently, in t h i s theory there i s no d i s t i n c t i o n between the quantity and quality of a dwelling unit as these terms are customarily used.... The assumption of a homogeneous good c a l l e d housing service can only be rejected i f theories of the housing market without t h i s assumption have greater explanatory power. The existence of rent control would of course v i o l a t e assumption (6). However the important issue i s indeed whether the kind of Kantian idealism expressed in assumption(7) (whereby housing service i s the noumenon of which housing type i s the phenomenon) does in fact represent the essential nature of the process of housing supply and demand. We begin by examining the nature of the supply curves in f i g . 8.1a. PVs'the most serious allegation against rent control i s the reduction of supply i t causes, through the a t t r i t i o n of the p r o f i t motive, i t i s important that the way the model determines the size of p r o f i t i s free from any d e f i c i e n c i e s . Mishan(l968, 1981) provides a 246 thorough discussion of t h i s problem, and his analysis i s followed here. The case against rent control made by the model graphed in f i g . 8.1 can be restated as being in effect are d i s t r i b u t i o n of income from landlord to tenant (Lipsey, 1971) the size of this r e d i s t r i b u t i o n may be expressed , according to t h i s model, as being the increase in consumer surplus over the loss of producer's surplus. Consumer surplus in the equilibrium case is the area between the demand curve and the supply price pJ^E • The actual price paid for housing i s p e and quantity of housing provided at that price i s Q^ s « Therefore the t o t a l rents paid is given by the area of the rectangle pe0Q^£. However out to the point Q j ^ r the amount of money consumers would be w i l l i n g to pay for that quantity of housing i s given by the trapezium Q D E Q j ^ . The area in excess of F ^ Q ^ S E (the amount actually paid) i . e . t^DE. thus represents the surplus accruing to the consumer by virtue of the fact that he or she pays less than they would be w i l l i n g to pay i f necessary. S i m i l a r i l y the area below the supply price and above the l i n e S^E i s regarded as being producer's surplus. A l l the return that is necessary to c a l l forth production up to the point is given by the area The fact that the producer receives in fact a renumeration equivalent to the area 0f> E -Q j ^ , means that the model, there w i l l also be a gain in consumer's surplus 247 equivalent to p c]^'£ C i a loss in producer's surplus equivalent to T^FG} , with the net gain in consumer's surplus over the loss in producer's surplus given by the tr i a n g l e £FQ. It is th i s gain in consumer's surplus over the loss in producer's surplus that represents the extent to which income i s transferred from the producer to the consumer by rent cont r o l . It is also that loss in producers' surplus or, that i s to say, p r o f i t s , which causes supply to f a l l . There are a number of objections to be made to th i s account of the p r i c i n g and supply of residential accommodation. In the short run with supply fixed, the substitution and income effects of a change in the price of oi housing i s in the natureyyeconomic rent. That is to say i t i s a payment to the supplier over and above that which i s required for him or her to maintain his or her investment in rental accommodation. So long therefore as the administered price i s above that which i s necessary to maintain the investment, there is no reason why the quality of the current housing stock should in any way suf f e r . As rent controls are generally imposed in order to prevent landlords from u n f a i r l y increasing their prices at a time of shortage, the administered price i s unlikely to have th i s e f f e c t . In the case of the long run supply curve, however, the application of the concept of producer's surplus i s i n v a l i d (in Mishan's words " i t is misleading and otiose"). This is due to the way in which the long run supply curve of micro— theory i s 248 constructed, We turn therefore to the long period supply curve for a competitive industry X in which, following standard textbook procedure, we assume a l l firms of equal size and e f f i c i e n c y . With unchanged techniques some i n e l a s t i c i t y in the supply of each factor, and an absence of external economies, the minimum long period average inclusive cost curve for each firm r i s e s as supply is expanded in. response to increasing demand. This r i s e in the industry's supply curve r e f l e c t s the growing scarcity of the factor that i s intensive to the product x in question. With only two factors c a p i t a l and [land], the production of an increased supply of x ent a i l s a r i s e in the price of c a p i t a l r e l a t i v e to [land], and owing to the greater r e l a t i v e of c a p i t a l i n x , a r i s e in i t s per unit cost r e l a t i v e to [land] intensive goods. With unchanged demand for an unchanged stock of money, this translates into a ri s e in the average money cost of X. (Mishan, 1968) Any point along the r i s i n g supply curve of good or good \j indicates the minimum average inclusive cost for each industry, and for each of the equally e f f i c i e n t firms engaged in i t s production. Thus in f i g . [8.2] at the market output of good , equal to Oat*, the minimum average cost 2 4 1 % i s the same for a l l of these firms and — i f perfect competition is assumed — equal to supply price 0p A . The long period envelope cost curve for the t y p i c a l firm, when long period equilibrium output Q*-± i s produced i s shown in the figure as SASA . And we may suppose that the Oxi output i s produced by such competitive firms. For a larger equilibrium output the minimum average inclusive cost i s again, the same for a l l firms and equal to supply price Op^ . We may suppose now that n firms (n being greater than m) in long period equilibrium are needed to produce th i s larger Oat-a output, the envelope curve for the t y p i c a l firm being Bx&x. • Since any movement . along t h i s long period industry supply curve S'B" i s to be regarded as a movement from one long period firm to another, there can be no net gains to the firms making the output x ... Each of the competitive firms i s making zero p r o f i t in long run equilibrium. (of course one of the factors could b e c a p i t a l instead of, or in addition to land. But the return to FIGURE 8.2: Long Period Supply Curve 250 c a p i t a l i s not p r o f i t to firm in the c l a s s i c a l Knightian d e f i n i t i o n , any more than is the return to land, or the return to labour. Knightian p r o f i t arises only in disequilibrium situations, a residual above a l l payments to factors and i s generally regarded as a return to enterprise, or as a reward for uncertainty bearing. In the context of comparative statics,„any point along the long period supply curve J9'S indicates the expenditure only on a l l the factors. Since the curve i s a locus of equilibrium points, there i s no Knightian p r o f i t , positive or negative, to induce firms to move into or out of the industry)... But i f t h i s long period supply curve i s an average curve including rents, can i t be regarded also as a marginal curve excluding rents?.... The short answer is no, not for a competitive industry... In the absence of the perfectly discriminating monopsonist, and in the real world ,this species of producer i s rather rare — the . S'S curve cannot be regarded as a marginal curve excluding rent, and no 'surplus' equal to the triangular shaded area can be associated with t h i s upward sloping supply curve and be regarded therefore as a transfer to the producers of payments otherwise made to factors. Certainly the shaded area cannot be taken to be the rent to one of the factors only (as, say, the rent of land in the Ricardian case). To conclude, insofar as the upward—sloping supply curve S'iS" of figure [&•?-] i s that of a competitive industry, i t i s not possible to regard the shaded area either as a 'producer surplus' or as a measure of rent to the factor—owner whose factor price rises as the good i s expanded. Such a competitive supply curve cannot be treated as more than an average curve including rents to a l l factors—which factors, we are to remember are d i s t r i b u t e d among a l l other industries (Mishan, 1981) In short, in conditions of long run equilibrium, the l e v e l of rentals has no effect on the p r o f i t a b i l i t y of the rental accommodation industry. Therefore a reduction in price from p& to pc cannot be associated with a loss in producer's surplus of , nor in the long run, with a transfer of income from producers to consumers. 251 Turning now to the demand curve employed in f i g . 8-^1 , r e c a l l the discussion of the inadmissability of applying comparisons of differences to analyses of change over time (£k67iffTJfe Sraffa System). To repeat, in a graph showing the interaction of supply and demand, time l i e s along an axis perpendicular to the plane of the axis. That being so, nothing in the supply demand re l a t i o n can be expected to remain constant. If supply grows more e l a s t i c i t y as the time period in question increases so also does demand grow more i n e l a s t i c i t y , c e t e r i s paribus. Yet in the model given in f i g . oVl, the demand curve i s the same regardless of whether the short or the long period i s considered (cf. Block, 1981, in Fraser I n s t i t u t e , 1981, p.311, f i g . 2). In a sit u a t i o n where demand i s perfectly i n e l a s t i c * % the p r o f i t a b i l i t y of supply unaffected by the price of supply, then regardless of what the price i s on these asumptions (long run s t a t i c equilibrium, with f u l f i l l e d expectations) supply w i l l always meet demand. Also, i f we r e c a l l the discussion of u t i l i t y theory, especially the assumptions necessary to construct an aggregate demand curve, then, i f th i s analysis i s to apply to more than one consumer at a time, the product, housing, and not merely i t s q u a l i t i e s , housing services, has to be treated as homogeneous. However, even i f one consumer is being considered, the idea that the c o l l e c t i o n of commodities among which he or she has to choose, i s heterogeneous, but that their q u a l i t i e s are i d e n t i c a l , i s a very contradictory one. Another problem with t h i s approach l i e s in the mechanism by. 252 which i t i s supposed to operate. If rents are permitted to ri s e in response to the point at which excess demand i s choked o f f , how then are signals to be transmitted to produce more housing? The normal method of estimating demand for rental accommodation is speed of response to the f i l l i n g of new vacanc ies.,7. "Demand i s to be understood as the flow of households who, in one unit of time, are looking for houses to purchase" (Charles, 1977). If the price i s set so as to equilibrate supply and demand in a situation of short run i n e l a s t i c i t y supply conditions, then should any vacancy occur for any reason, i t w i l l not be f i l l e d u n t i l i t s price drops below the equilibrium p r i c e . In such a situation therefore, high prices w i l l not encourage the production of new stock. The immediately forgoing c r i t i c i s m depends on the assumption that the role of response to prices to a (single) r i s e in demand is immediate. I f , however, changes in price occur at a slower rate, the r i s e in prices may well act as a signal to bring forth new construction during the period between the beginning of the ri s e s in price and the point at which they choke off demand. If there are also lags in supply, supply may well overshoot demand as housing, begun when prices f i r s t began r i s i n g , arrives on the market after prices have reached the 'equilibrium' l e v e l . Since suppliers w i l l not know what thi s l e v e l i s when they start construction, they w i l l l i k e l y build to a standard commensurate with the price expected at the end of the construction period, an expectation based on the current 253 continuing. Depending on the r e l a t i v e speed of movement of prices toward T>e as compared with the speed of construction of new dwellings, the average prices of new construction w i l l tend to be higher or lower than p& , higher when the rate of price response i s faster than the rate of construction, and lower when vice versa. Only in the case where the rate of price response is slower than the rate of construction w i l l the changes in price and supply lead to an increase in stock which does not overshoot the r i s e in demand. This i s unlikely to be the general case, however, unless some form of price controls have been i n s t i t u t e d (such as rent control ) f l ,,as the headline story "Landlords Hunt Tenants As Vacancy Rate Rises" (Vancouver Province, 1 March, 1982) indicates. However, in .the model as presented in F i g . 8.1, no informaton is available that might indicate the presence of such lags in supply or price change. Before we f i n a l l y abandon th i s model, however, we need to make one more c r i t i c i s m of the assertion that the reduction of price leads to reduction in supply. Even on the basis of Fig. 8.1, t h i s assertion cannot be maintained. So long as the administered price y i e l d s an economic rent to the landlord, there w i l l be entry into the construction of rental accommodation (with the attainment of long—run equilibrium with zero p r o f i t s , t h i s movement w i l l cease). What the price w i l l a f f e c t , however, i s the rate of entry and hence the amount of construction of new dwellings. So long as the administered price f a l l s between the equilibrium price and the minimum necessary to 254 y i e l d an economic rent. It w i l l a f f e c t the rate at which adjustment to a new equilibrium takes place, but i t w i l l not affe c t the f i n a l outcome. This point i s worth stressing, for i t i s frequently neglected in discussions about rent control. S p e c i f i c a l l y , in terms of [ F i g . 8.1b ], the f i r s t year's addition to the stock w i l l be 0c\a instead of Oc|.e , and in each subsequent year, the annual increment w i l l be s i m i l a r l y smaller. But a stock y i e l d i n g (ty e w i l l be achieved eventually. Thus, in such cases shortages of housing can be said to aris e because of rent control insofar as the amount of new construction is lower than i t otherwise would be, but i t should not lead to la s t i n g shortages (Robinson, R.V.F, 1979, p. 77). Although the discussion of the micro—economic analysis of the housing market has been conducted with reference to the debate over rent control, c l e a r l y the c r i t i c i s m s applied in th i s case apply to housing supply and demand in general. The same model as used in this case apply to housing supply and demand in general. The same model as used in the case of private rental housing i s also applicable to the case of housing in general (Robinson, R., 1979). We return , therefore, to a discussion of the a b i l i t y of micro—economic analysis to derive a re l a t i o n between prices and quantities in general. 255 8.3 The Keynesian Critique Of Micro—Economic Price Theory In the previous section we argued that there was a paradox in the orthodox micro—economic account of the mechanism by which price changes were supposed to equilibrate supply and demand. If the response of prices to changes in the l e v e l of demand were instantaneous (and there was no reason to suppose they were not) then an increase in price in response to an increase in demand would c a l l forth no new supply since the effect of the price increases would be to choke off excess demand. If there were no excess demand at the new higher price, there would be no new supply. Furthermore, in the long run, supply would always adapt to meet demand regardless of the l e v e l at which the price.of new supply might be set. The implications of t h i s argument are that i f new supply i s not affected by movements in prices, then prices cannot be regarded as a determining influence on quantities offered. A l l they can be used for then would be as a way of confirming whether the market is in equilibrium, but not why i t i s (or i s not) (Kregel, 1973). It was argued that the sources of t h i s paradox were therefore to be found in the construction of the supply curve. However the refinements to the model which would this entailed could not be accommodated within the structure of the orthodox theory, as i t was presented. To avoid the paradox described above, but yet maintain the structure of the orthodox theory, requires the assumption that buyers and s e l l e r s may exchange information on prices, but without actually carrying out any process of exchange. Haggling 256 is a one to one example of t h i s . The general case requires the presence of a Walrasian auctioneer whose job i t i s to announce a l l bids and offers (in order that a supply and demand schedule may be established) before permitting trading to commence. The Walras type economy was b r i e f l y discussed in the Introduction, where the implications of the existence of economies of scale were shown to be inimical to the assumption of perfect foresight, which in turn implied the need for l i q u i d i t y and therefore the existence of a monetary economy. However even i f we grant constant returns, i t can be shown that the problem of pricng the quantities supplied and demanded such as discussed in the previous section, is not one s p e c i f i c to the housing sector, but one which has f a t a l implications for micro—economic theory in general. We begin with a brief account of the Walrasian system in order to set the stage for the discussion of the problem of re l a t i o n prices and output that follows. This section r e l i e s heavily on Kregel, 1973, chs 1&2 . In Walras' system the d i s t r i b u t i o n of the physical production i s given in advance and the groping [tatonnement] for the market price that equates supply and demand is something that only happens in imaginary or l o g i c a l time. There is no way to handle the production of goods over time nor the movement through history that t h i s requires prices are determined instantaneously and c o s t l e s s l y and provide a l l traders a l l the information that they need to know to make their decisions, also without cost or elapsed time. But there i s l i t t l e point in c r i t i c i s i n g t h i s story (or Edgeworth's process of recontracting) because i t i s not meant to be r e a l i s t i c . It i s only supposed to be a story that reproduces the results the free market i s presumed to produce in i t s own 257 mysterious and i n v i s i b l e way. The point we should note however i s that this story in no way serves as a proof that the market r e a l l y functions in the manner assumed, i t may be true or false without being an accurate description of the workings of the market. Before and after Walras' infamous auctioneer, the i n v i s i b l e hand remains l i k e most r e l i g i o u s apparitions, only seen i f believed. But i t is on t h i s story that the neo-classical theory of market prices rests. (Kregel op.cit., p. 13-14) The auctioneer guarantees that the system w i l l always be in equilibrium by ensuring that the requisite prices are in force before trading begins, yielding a " p r e — r e c o n c i l i a t i o n " (Shackle, 1972) of buyers and s e l l e r s in "pre-time" (Kregel, 1973). The circumstances which w i l l a ffect the outcome of given conduct include the actions of other individuals in order that each person may choose his conduct in e f f e c t i v e knowledge of the contemporaneous choices of others, there must be a p r e — r e c o n c i l i a t i o n of a l l choices by means of a declaration and pooling of conditional intentions. Equilibrium i s the prescription of each person's conduct in accordance with his own preferences thus declared which emerges from that pooling, i f such e x i s t s . . . The formal notion of pre—reconciled choice solves, within a highly a r t i f i c i a l and abstract frame, the problem of how men make the i r choices of action in f u l l knowledge of each other's simultaneous choices. Such solution i s necessary to the notion of s t r i c t r a t i o n a l i t y , the exercise of reason upon complete relevant information, since the l a t t e r must include the intended actions of others. (Shackle, 1972) p. 53-4, p. 104) The need for an auctioneer, or some similar concept to achieve such p r e — r e c o n c i l i a t i o n i s apparent from these quotations. It i s clear also that the notion of p r e — r e c o n c i l i a t i o n i s closely linked to that of Rational 'Economic Man. In the absence of an auctioneer, p r e - r e c o n c i l i a t i o n i s assumed, not proven, without i t , there i s 258 no pressure on the system to ever achieve equilibrium. However, as we have seen, these assumptions were only made in order to construct a theory of what the economists of the time t h o u g h t the market did. The market was assumed to equate supply and demand and, in addition (or possibly as a reassuring bonus) to give optimal d i s t r i b u t i o n and a l l o c a t i o n of resources by providing perfect information. (Kregel, op.cit., p. 13) Even the notion of economics as 'the study of the optimum a l l o c a t i o n of scarce resources among alternative ends' depends on th i s construction. For i f the d i s t r i b u t i o n of the physical product i s given in advance, and the process of production not considered so that they are not able to be employed in increasing the amount of output produced, then i f resources have any price at a l l , i t is because of their s c a r c i t y , (as we have seen e a r l i e r t h i s position cannot be maintained under conditions of increasing returns, Kaldor, 1972) Walrasian economics i s , as Shackle (1972) argues, a theory of r a t i o n a l conduct, and one therefore confined to a timeless or momentary world. The questions as to i t s a p p l i c a b i l i t y to the real world of h i s t o r i c a l time are magnified therefore when i t is used as the basis of dynamic analysis. It i s in these applications, Kregel writes that one can see how inessential t h i s theory i s , for these models seem to have to behave according to market p r i n c i p l e s even when there is only one produced commodity and hence l i t t l e role for r e l a t i o n prices to play. In this approach the decision to save determines both investment and the real wage required to produce f u l l employment over time. In such a system the story of the auctioneer is hardly worth t e l l i n g , for he has almost nothing to do. (Kregel, op.cit.) 259 Pasinetti (1974) makes a similar point when he writes Solow, for example, after writing that his work is devoted to a model of long run growth which accepts a l l the Harrod—Domar assumptions except that of fixed proportions b l i t h e l y goes on to add a whole series of other assumptions which Harrod and Domar do not make: a d i f f e r e n t i a b l e , linear and homogeneous production function, perfect and i n f i n i t e s u b s t i t u a b i l i t y of labour and c a p i t a l perfect competition in the labour and c a p i t a l markets, etc. Moreover he makes the formidable assumption that there i s only one commodity in the whole economic system. The extraordinary feature of these assumptions is that they are not only numerous, but p e c u l i a r l y hybrid opposite and extreme. On the one side the existence i s assumed of only o n e commodity and on the other side the existence is assumed of an i n f i n i t e number of techniques for producing i t . On the one side the Keynesian view i s accepted that the rate of interest has no importance in determining savings, and on the other side the opposite assumption is made that the rate of interest is so important as to make the capital—output r a t i o vary from near zero to near i n f i n i t y . It is hard to see any rationale for this set of assumptions except that of bringing in a p a r t i c u l a r shape of technology—and thus marginal p r o d u c t i v i t y — a t a l l costs, as the main determinant of income d i s t r i b u t i o n and the rate of p r o f i t . Although the auctioneer was supposed to set the prices, i t is the prices themselves that were supposed to provide the information necessary to effect trades to the mutual s a t i s f a c t i o n of a l l agents in the system. This mutually sati s f a c t o r y outcome of the trades i s also the one which ensures the optimal and complete a l l o c a t i o n of a l l resources. At t h i s point. The terms of reference of the theory are widened to analyse what actually happens in actual economy. If resources are priced at a l e v e l which ensures their f u l l u t i l i s a t i o n , then 2 6 0 the only way f u l l u t i l i s a t i o n can f a i l to occur i s i f the owners of those resources hold out for a higher than equilibrium price for them. In such a s i t u a t i o n , the system w i l l not only be thrown into disequilibrium, but those factors which have been too highly priced w i l l remain unemployed. This i s the basis of the (neo—) 'Classical Theory of Employment' that Keynes destroyed. In this case the price of labour is regarded as being too high, and the cost this represents to empolyers drives down their p r o f i t a b i l i t y and force them to reduce the number of jobs they can supply. In the case of rental housing, the price of renting to households i s too high (this i s equivalent to the cost of l a b o u r — o f h i r i n g labour—being too high; r e c a l l that one argument against rent control i s that i t represents an 'arbitrary' r e d i s t r i b u t i o n of income from landlords to tenants, equivalent to the r e d i s t r i b u t i o n of income from p r o f i t s towages forced on employers by wages being too high), t h i s reduces their p r o f i t a b i l i t y and forces them to reduce the number of apartments they can supply. The p a r a l l e l s between the two cases are quite e x p l i c i t and as s h a l l be seen, Keynes' c r i t i q u e of the ' C l a s s i c a l Theory of Employment' as i t applied to the employment (or lack thereof) of labour can be applied equally to a c r i t i q u e of the c r i t i q u e of rent control, and to the problem of housing shortages in general. While we have outlined the general p r i n c i p l e s upon which the Keynesian theory i s b u i l t and how the post-Keynesian theory develops out of t h i s , we have yet to review Keynes' c r i t i q u e of 261 the t r a d i t i o n a l ' C l a s s i c a l Theory of Employment'. There two strands to his c r i t i q u e . The f i r s t i s an e x p l i c i t denial of the Second C l a s s i c a l Postulate, as Keynes termed i t , that the money wage, giving the labour supply schedule, is determined by the marginal productivity of labour. If thi s were true, then labour, being in a position to decide i t s marginal d i s u t i l i t y , would be responsible for any unemployment i t may suffer through i n s i s t i n g on a money wage that was higher than the real wage employers were able to o f f e r . The real wage, or labour demand schedule i s yielded by the marginal productivity of labour, which i s the thrust of the F i r s t C l a s s i c a l Postulate. Though Keynes is w i l l i n g for the sake of argument to accept the F i r s t C l a s s i c a l Postulate in order to concentrate his attack on the second, this i s in fact subject to an equalling damaging but im p l i c i t c r i t i q u e . By arguing that job supply depends on the level of investment, the argument that technological conditions (marginal productivity) has a determining ef f e c t on job supply i s thereby refuted. The attack on neo-classical economics mounted by the post—Keynesians has been in the main a working out of this i m p l i c i t c r i t i q u e . This has led to a tendency to overlook Keynes's own e x p l i c i t c r i t i q u e , but thi s is one which is s t i l l equally powerful as a means of rebuttal of of neo-classical theory. The p r i n c i p a l point of Keynes's c r i t i q u e was that the price system of neo-classical theory provided i n s u f f i c i e n t information to be able to ever y i e l d a set (or vector) of market clearing 262 prices for a l l goods. The C l a s s i c a l Theory of Employment "was p i l l a r e d on two p r i n c i p l e s : Say's Law and the Quantity Theory of Money" (Kregel, op.cit. ) Say's Law as understood by the neo-classicals was that supply creates i t own demand, which would be achieved via adjustments in the r e l a t i o n prices of goods. If there were excess supply of a factor, i t s price would be driven down and i t would be substituted into the productive process in place of other, dearer, factors. In p r i n c i p l e therefore, there would always be a "set of relative, prices which eliminated excess supply or demand and thus generated the f u l l employment of a l l factors traded on the market in competitive conditions" (Kregel, o p . c i t . ) . Implicit in this account was the assumption of perfect s u b s t i t u t a b i l i t y between a l l factors, including therefore the p r i n c i p l e of homogeneous malleable c a p i t a l . In addition i t i s necessary to assume " s t r i c t independence of micro—economic it supply and demand functions for a l l factors traded. Savings are also assumed to be linked to investment via the rate of interest, which since the rate of interest i s assumed equal to the rate of p r o f i t , also gives the price of c a p i t a l . An increase in savings brings about a f a l l in the rate of interest which also cheapens the price of c a p i t a l goods re l a t i o n to consumption goods, and thus stimulates investment. On the consumer side a l l income and price e l a s t i c i t i e s of demand are assumed equal to one. Although t h i s gives the relative prices in terms of 263 p h y s i c a l product r a t i o s , the ab s o l u t e p r i c e l e v e l i s undetermined. T h i s i s where the Quantity Theory of Money comes i n . The b a s i c i d e n t i t y i n v o l v i n g money i n the economy i s M = PT The money supply (rt) times the v e l o c i t y of c i r c u l a t i o n (V) i s at a l l times the same as the p r i c e l e v e l (P) times the volume of t r a n s a c t i o n s ( T ). T h i s r e l a t i o n i s r e a l l y no more than an ex post accounting i d e n t i t y , but the Quantity T h e o r y turns i t i n t o f u n c t i o n a l r e l a t i o n by w r i t i n g P = f -\J i s c o n s i d e r e d constant, and T i s given by technology and the number of agents i n the economy. Thus the money supply determines the l e v e l of money p r i c e s without however a f f e c t i n g the r e a l exchange r a t i o s measured i n terms of p h y s i c a l p r o d u c t 9 "thus both r e l a t i o n p r i c e s and the ab s o l u t e p r i c e l e v e l were e x p l a i n e d . The system was n e a t l y c l o s e d but a l s o n e a t l y separated." (Kregel, o p . c i t . ). Money was j u s t a " v e i l " , an unnecessary c o m p l i c a t i o n to be c l e a r e d away before a n a l y s i s of the ' r e a l ' v a r i a b l e s i n the system c o u l d begin. However what c o u l d never be adequately e x p l a i n e d was how changes i n the gene r a l nominal p r i c e l e v e l governed by MY= P X ' r e l a t e d to changes i n the p r i c e s of i n d i v i d u a l products set by cost and demand c o n d i t i o n s at the micro—economic l e v e l . Keynes' T r e a t i s e on Money was an attempt to provide a theory which would l i n k the det e r m i n a t i o n of i n d i v i d u a l p r i c e s with the o v e r a l l p r i c e l e v e l . T h i s he d i d by showing that the r e l a t i o n of savings to 264 investment could affect the general price l e v e l , and that therefore "real and monetary, phenomena were inextricably linked in an economy with uncertainty ever possible outcomes of future events, i . e . , in what he c a l l e d a monetary economy" (Kregel, op.c i t . ) . Even so, the Treatise s t i l l subscribed to the view that unemployment was due to disequilibrium and could be cured by the necessary adjustments in r e l a t i v e prices, prices which were out of equilibrium due to a "poorly conceived [central bank] p o l i c i y or the existence of an unstable banking system." (Minsky, 1975). The demonstration that the problem of unemployment was a result of the inadequacies of the price system was not made u n t i l the publication of the General Theory of Employment Interest and Money. As we have seen, the C l a s s i c a l Theory of Employment, with i t s assumptions of independence of micro—economic supply and demand functions and of perfect substitution between factors held that unemployment of factors could only be the result of vo l u n t a r i l y witholding those factors from the market at the rul i n g price, that is holding for higher prices than the market can bear. The solution offered by the C l a s s i c a l theory i s that therefore that labour must cut i t s wages to increase employment. If the labour market were to operate e f f i c i e n t l y howver the result i s reminiscent of our analysis of the effects of removing rent controls. Moreover the implications are devastating to the proposition that the price system can act as a s i g n a l l i n g 265 mechanism for the a l l o c a t i o n of resources, and the determination of quantities of output. Kregel writes: Now the best result which could occur in the case of decreasing money wages would be no change at a l l in the amount of employment of fered which could conceptually occur i f prices and wages adjusted instantaneously to the new conditions. In such a case there would be no change in relation prices or employment. The process that the neo-classical explanation required, a f a l l in wages r e l a t i o n to prices could not be produced without the assumption that aggregate demand was independent of the l e v e l of wages or that f u l l employment was guaranteed from outside the system. The net result of the proposotion then must be that changes in r e l a t i o n prices would not (indeed could not) produce the advertised r e s u l t s — t h e f u l l employment of a l l factors. The operation of the price mechanism to produce f u l l employment f a i l e d in the c r u c i a l test on the labour market. If the theory is not applicable to the factor labour, there appears to be no reason why i t w i l l work in any other case. F u l l employment was something that was outside the control of r e l a t i o n prices. (Kregel, op.cit., emphasis added) To elaborate on the argument that changes in r e l a t i o n prices could not produce the advertised results of f u l l employment of a l l factors, note that i f aggregate demand i s indeed independent of the l e v e l of wages, then what i s being said i s that labourhs purchasing power over ( i . e . , demand for) a l l other products is not affected by the price of labour ( i . e . , the l e v e l of wages). The p r i n c i p l e of e f f e c t i v e demand, and i t s d i s t i n c t i o n between .existing and employed capacity was the basis of Keynes' refutation of this assumption. A l t e r n a t i v e l y i f we have f u l l employment guaranteed from outside the system, then i t i s also guaranteed from outside the realm of influence of r e l a t i o n p r i c e s . That the system of r e l a t i o n prices f a i l s to 266 ensure f u l l employment is a point also made by Nuti (1970b) quoted e a r l i e r who said that i t was a necessary condition for the p o s s i b i l i t y of a perfect labour market in the present period, that there can be no futures markets in labour. Thus attempts to make the Walrasian general equilibrium approach dynamic, by holding the price system to account for the intertemporal a l l o c a t i o n of production and consumption, must therefore always f a i l , since in the c r u c i a l question of the al l o c a t i o n of labour, there can never be a set of prices which determines i t s a l l o c a t i o n over time. Thus the problem with the neo-classical account of the price system as an a l l o c a t i v e device may be said to be that i t cannot deal with expectations and i n t e n t i o n a l i t y . As Shackle says, the world of equilibrium economics is a momentary one, and i t s demand and supply schedules momentary also. For they depend upon expectations about the future, and are therefore i n f l i c t e d with the essential uncertainty that- this implies. Even i f individuals responded to changes in r e l a t i v e prices, the s h i f t s in the supply and demand schedules that these price changes would incur, by reason of the changes in expectations that they would engender would far outweigh the effects of any changes due to movements along the supply and demand curves. A good example of the i n a b i l i t y of the price system to capture the effects of i n t e n t i o n a l i t y and expectations i s provided by the-housing market. Micro—economic price theory has i t that i f prices r i s e , demand f a l l s . However i f house prices 267 r i s e , t h i s often stimulates demand as people attempt to get 'on board' before prices r i s e too high for them to afford, or in the prospect of being able to r e a l i s e a p r o f i t in their intended investment. In general, since investment means committing a current income to the hope of future income, the influence of expectations and uncertainty w i l l weigh heavily upon the decision to invest. Also changes in expectations about prospects for future income on the part of consumers may lead to "precautionary balances being b u i l t up at the expense of current consumption. Changes in future income may then also cause a s h i f t in the consumption function" The co-ordination of decisions to invest and decisions to save at a l e v e l that produced f u l l employment of labour was a problem that the price system alone could not handle for i t could niether provide the necessary information, nor produce effects on expectations that might overcome th i s lack of information." (Kregel, op.cit., p. 11) That the price system cannot handle such problems should come as no surprise once i t i s realised that neo-classical economics superimposes the relations c h a r a c t e r i s t i c of an economy based on barter and exchange, where money i s only useful as a unit of account, onto the features of an economy based on i n d u s t r i a l production, where money i s not only a unit of account but also a means of dealing with uncertainty about the future, a l l a y i n g fears about future prospects because of i t s powers of l i q u i d i t y , powers which derive however only from people's estimation of money's a b i l i t y to have this property, to behave 268 in t h i s way (Shackle, 1972). It is a convention only, and one which can be overthrown, as for example in post—war Europe, where cigarettes and chocolate were more important as money than the contemporary currencies. 8.4 E f f e c t i v e Demand And The Provision Of Housing: The Short Run Thus far we have i d e n t i f i e d the orthodox account of housing production and the c r i t i q u e of rent control that i t gives r i s e to with standard microeconomic theory. We have seen how the v a l i d i t y of th i s theory stands and f a l l s with the a b i l i t y of the theory to guarantee f u l l employment of a l l factors of production. The fact that i t cannot explain the unemployment of labour renders i t useless to explain the employment of any factor. A l l that remains now therefore i s to demonstrate how the housing sector can suffer from Keynesian - type d e f i c i e n c i e s in aggregate demand. As we have seen, unemployment in the Keynesian system arises because decisions to save and decisions to invest are made for d i f f e r e n t reasons and the price system i s unable to co-ordinate these two aspects of the production process. S p e c i f i c a l l y , transferring income from labour to c a p i t a l , that is to say a l t e r i n g the r e l a t i v e prices of labour and c a p i t a l , by cutting money wages w i l l not solve the problem of unemployment o 269 of labour. Shackle (1972) puts the problem in these terms When the wage demanded and accepted by suppliers ov productive services i s expressed wholly in money, in general non-specific purchasing power, and that which i s paid by the employers is expressed partly in concrete, specialised and durable equipment, that a great gulf can open up between the composition of the wage (or other pay) desired by the income—earners in terms of consumption or savings, and the composition of the pay which the employers are w i l l i n g to o f f e r , in terms of consumable goods or investment goods, at a l e v e l of aggregate general output and income corresponding to f u l l employment, so that f u l l employment becomes impossible. The discrepancy between offered and demanded product i s not one of size but composition. Wage—earners, or rather the whole body of suppliers of productive services including even the business men themselves in their capacity as income—receivers and disposers, rather than as enterprisers and employers desire a wage or other pay, which in product terms consist partly in provision for the future rather than enjoyment in the present. But at certain conjunctures in their a f f a i r s at certain h i s t o r i c a l epochs, employers are unwilling to offer a proxy for the income—earners and act as the nominal de jure owners of the real povision for the future, the i n d u s t r i a l f a c i l i t i e s and equipment, which alone enrich society as a whole, while the real owners, the income—earners who had made possible the accumulation of those tools by saving part of their incomes, merely lent the saved money to the employers. At times of depressed business or uncertain p o l i t i c s , at times of threatened p o l i t i c a l change, employers do not want to become debtors in money in order to gamble on the fortunes or misfortunes of technically specialised, invent ion—vulnerable, market—dependent concrete equipment [there can be] a non—compatability between the 'mix' of consumption and wealth—accretion desired at f u l l employment by the income—earners, that i s to say, the suppliers of productive services of a l l sorts, to whom the produce of industry as a whole necessarily belongs, and the 'mix' which the businessmen, in their 270 capacity as such, the risk—bearers of i n d u s t r i a l processes which are technologically compelled to look years ahead for their consumnation and reward, are w i l l i n g to give. For the income—earners save money which they lend in one way or another back to the businessmen... But money for sociey as a whole, for the body of income—earners a l l taken to—gether, i s no use as a provision for the future; only equipment, real tools and f a c i l i t i e s can constitute such provision. However, these tools and f a c i l i t i e s w i l l have to be formally and l e g a l l y the property of the businessmen. The businessmen, when they invest with borrowed money, become hostages to an incalculable technological and fashionable future. The extent to which they are w i l l i n g to do so may not match the extent to which income—earners... wish to save out of the aggregate income corresponding to f u l l employment. In times of uncertainty, entrepreneurs would rather hold their assets in l i q u i d form, as money. Money i s desired since i t represents a means of escape from the hazards of ownership of specialised and c r y s t a l l i s e d forms of wealth, exposed as they are constantly to obsolescence and loss of value. Money represents general purchasing power, a form of wealth which enables a man to put off making up his mind what his savings s h a l l buy for him; to put i t o f f , possibly, for ever. (Shackle, 1972, ch. 17) A similar interpretation i s put forward by Minsky(1975). For Minsky, the General Theory i s one in which the causes of fluctuations in investment are changes in desired private p o r t f o l i o composition of economic units, especially firms and households, but also banks and other f i n a n c i a l i n s t i t u t i o n s . A p o r t f o l i o which consists of assets owned or controlled, and l i a b i l i t i e s put out to achieve t h i s ownership and control, involves the existence of decision units in a present position which r e f l e c t s current and past views 271 about the prospects of p a r t i c u l a r units, as well as of the economy. (Minsky, op.cit., p. 69) The effects of uncertainty upon desired p o r t f o l i o s and of evolving p o r t f o l i o s upon d e s i r i r e d p o r t f o l i o s can be such that the equilibrium to—wards which the system tends not only i s always changing, but can change rapidly. Thus the behaviour of the economy i s characterised by e q u i l i b r a t i n g tendencies rather than by any achieved equilibrium. Keynesian economics as the economics of disequilibrium is the economics of permanent disequilibrium, (op.cit., p68) In Keynes' theory the proximate cause of the transitory nature of each c y c l i c a l state i s the i n s t a b i l i t y of investment, but the deeper cause of business cycles in an economy with the f i n a n c i a l i n s t i t u t i o n s of capitalism i s the i n s t a b i l i t y of p o r t f o l i o s and of f i n a n c i a l i n t e r r e l a t i o n s . . . Keynes put forth a theory of fluctuations in real demand, and a f i n a n c i a l theory of fluctuations in real investment, (op.cit., p. 57) I have quoted extensively here because the interpretation placed on Keynes' work by those two authors l i e s somewhat outside of the mainstream of post-Keynesian work. However i t i s important to emphasise that i t is f u l l y consistent with i t . The reason for the differences is that most post-Keynesian work has concentrated on demonstrating that Keynesian economics i s incompatible with marginal productivity theory, i . e . , on making e x p l i c i t Keynes' i m p l i c i t c r i t i q u e of the f i r s t c l a s s i c a l postulate. Keynes' neglect of the implications of his c r i t i q u e of the second c l a s s i c a l postulate for orthodox neo-classical micro—economic price theory by and large has not been attended to by his followers. Kregel, Shackle and Minsky, especially the l a s t named are among the few who have examined these implications. 272 The main outline of the argument may now be presented. Housing i s a consumption good, part of the product wage required by labour. Housing service is provided by investment in housing stock, something which investors are cautious about doing, since then, while their debts would be in money, their assets would be in c a p i t a l of the most fixed kind, b u i l t structures with l i t t l e opportunity of alternative uses. The existence of owner—occupation, where consumers are spoken of as having 'invested' in housing should not be allowed to obscure t h i s picture, since the sources of funds for this type of investment are mortgage finance i n s t i t u t i o n s . The a s s e t / l i a b i l i t y structures in the case of these i n s t i t u t i o n s causes the problem of uncertainty generally borne by entrepreneurs to be greatly magnified. These i n s t i t u t i o n s a t t r a c t finance on the basis of the guarantee that funds can e a s i l y be withdrawn. They compete with other f i n a n c i a l i n s t i t i u t i o n s for savers' funds on this basis. However every savings account opened with a mortgage finance i n s t i t u t i o n represents an increase in the i n s t i t u t i o n ' s l i a b i l i t y . Conversely i t holds i t s assets in the form of mortgages on houses thus their l i a b i l i t i e s are l i q u i d and their assets i l l i q u i d . Moreover the terms on which a mortgage i s granted, generally anywhere from 5 to 25 years, depending on the practices in a country or type of i n s t i t u t i o n mean that these i n s t i t u t i o n s are in the position of borrowing short and lending long (Charles, 1977). A second factor influencing the operation of mortgage 273 finance i s the necessity to maintain a reputation for soundness, of security of savings. This makes mortgage companies cautious in deciding to whom they w i l l make funds available and for what type of house. F i n a l l y , competition for savers' funds takes place through variations in the rate of return offered to savers. This operates in the reverse way to the second factor, too high a rate jeopardising the both the reputation and the r e a l i t y of security of savings as an increase in the rate of interest offered savers must be financed out of an increase in the mortgage rates charged to home purchasers. Too frequent changes in the rate of interest w i l l make the cost of homeownership uncertain for borrowers, and increases in the rate may cause defaults in existing mortgages. Thus there is some pressure to keep mortgage rates stable compared to the general interest rate. Should the general rate r i s e with respect to the mortgage rate, "the flow of funds f a l l s and less money must be rationed out over those who demand finance by s t r i c t e r operation of the lending rules; thus there is less e f f e c t i v e demand for houses." (Charles, op.cit., p. 42) This ."'-analysis of mortgage finance i n s t i t u t i o n a l behaviour i s not p a r t i c u l a r l y exceptional, indeed i t i s taken from a standard textbook on housing economics, by Susan Charles (1977). What i s d i f f e r e n t however i s the context within which th i s behaviour i s viewed. Thus Charles views mortgage companies' behaviour as anomalous when compared to normal market behaviour, 274 as can be seen from her discussion of the mortgage rate. The mortgage rate i s the price in the market for finance of house purchase. It i s not however a market determined price: i t is a price determined administratively... The le v e l of the rate i s set on a cost—plus basis. The rate on shares and deposits is dictated by conditions in the f i n a n c i a l markets. Once this rate is set, the mortgage rate follows as this rate plus a margin to cover management costs, taxation and additions necessary to maintain t h e i r reserve r a t i o . Thus, this price does not perform the usual function of a l l o c a t i o n , i . e . , i t does not a l t e r to bring about equilibrium in the demand and supply of mortgages. Instead a l l o c a t i o n i s by administrative decision on the part of [mortgage finance i n s t i t u t i o n s ] . They alloca t e supply over demand according to their assessment of the borrower's security. They w i l l vary the stringency with which they assess security as, demand and the supply of mortgages varies. (Charles, op.cit., p. 43) On the basis of the interpretation proposed here, Charles'account of the operation of the mortgage market i s exactly as one would expect. The interest rate equates the supply and demand for money, 0(L,H) i- > the mortgage company decides upon investment (here, what investment w i l l be permitted) over the set of possible opportunities, E , given the rate of interest, so that 1->"^"^EJL') X ' a n <^ the whole operation of th i s process gives the e f f e c t i v e demand for housing. Charles on the other hand, in the quotation given above writes as though the rate of interest was supposed to equate savings and investment (supply and demand of mortgages). The whole character of th i s process i s one which i s l i a b l e to chronic underinvestment in housing, both in the short and long term. The cause of short run underinvestment follows d i r c t l y from the account of the causes of labour unemployment given by 275 Shackle quoted above, that i s , of a discrepancy in the compositions of the desired and offered product wages because of uncertainty about the future, except that in t h i s case i t i s the le v e l of housing vacancies that f a l l s , not the l e v e l of job vacancies. (though this i s in no way to suppose that the two level s w i l l not be related). To paraphrase Shackle, the times w i l l always uncertain, by d e f i n i t i o n , and therefore there is no l i k e l i h o o d of a r e c o n c i l i a t i o n ever occurring, except by chance, of the expectations and desires of income—earners and employers (as income—providers), and thus of a product wage whose composition s a t i s f i e s both sides. The short run situation may also be presented in terms of induced and autonomous investment. If induced investment i s that investment which goes to—ward providing the wage good, and a known proportion of that goes on housing (in p r i n c i p l e measurable from the income and price economies of demand for housing) then we can sketch out' a diagram r e l a t i n g the induced demand for housing to various levels of labour—force income, (which covers changes due to increases in employment and real wages). This r e l a t i o n is graphed in f i g . 8.3. We make a linear approximation to V-LA = " + , where k i s the marginal . . . o propensity to invest in housing. The 45 l i n e shows how demand for housing is always equal to t o t a l income available for housing. Note that this r e l a t i o n w i l l be linear even i f the r e l a t i o n of Y k to t o t a l income i s not. Provision of housing w i l l be c a r r i e d out according to the formula for investment in 276 277 housing, e q u i l i b r a t i n g supply and demand at the point where Xj crosses the 45° degree l i n e , at ( Y^, D ). However the desired level of housing is given by the point ). The difference Y ^ - m e a s u r e s the difference between potential and actual provision of housing services. This is equivalent to,.©.-]) , therefore i t also represents the lack of e f f e c t i v e demand for housing. It is clear from the diagram that no amount of income transfers to the demanders of housing w i l l a ffect this s i t u a t i o n of lack of e f f e c t i v e demand for housing, in the absence of any increase in productive capacity in this sector. Unless, therefore, autonomous investment i s carried out to a l e v e l which makes up the difference between Y^ a n <3 '-'"^r there i s always l i k e l y to be a housing shortage, (the neo-classical solution i s to raise house prices, coupled with an income supplement, Fraser Institute, 1981. This could indeed have the effect of r a i s i n g the l e v e l of investment, l e t us say to Xy+X^f D U t in nominal terms only. When the supplemental additions to income are taken into account, i t i s clear that t h i s 'solution' only reproduces the problem at an i n f l a t e d l e v e l . The causes of the f a i l u r e s of such government programmes as AHOP may be recognised as lyi n g in this 'solution'.) At this point a couple of issues should be dealt with. Keynes' analysis develops out of a c r i t i q u e of Marshall, who used the method of p a r t i a l rather than general equilibrium. To what extent therefore i s i t f a i r to cred i t Keynes with having exposed the de f i c i e n c i e s of general equilibrium analysis? To 278 what extent moreover i s i t possible to talk about e f f e c t i v e demand other than in the context of demand for labour? Shackle deals with the f i r s t issue in a discussion of the conditions necessary for the mutual independence of demand and supply curves. The mutual independence of supply and demand conditions is only plausible i f we are speaking of a minute portion of the economy; only that i s for p a r t i c u l a r or partia1 equilibrium... To treat the labour market as being amenable to analysis by the apparatus of intersecting, mutually independent demand and supply curves, was p l a i n l y to abuse the device of p a r t i c u l a r equilibrium. The labour market is not a 'particular' market, but an e s s e n t i a l l y general one, encompassing a l l production. This does not mean that i t i s a perfect market. It is an assemblage of inter—acting markets. To scale up the method of p a r t i c u l a r equilibrium as a means of analysing as vast an aspect of things as the labour market i s of course to be g u i l t y of the ' f a l l a c y of composition' (shackle, 1972, p. 261) However p a r t i a l equilibrium i s not free from internal inconsistency either. In p a r t i a l analysis the firm i s a price taker, i t assumes the price of the product as given. However i t also regards t h i s market price as the outcome of the interactions of the exploratory conduct of firms. The market price a r i s e s from what firms do. They cannot know what to do, except by t r i a l and error, by seeing what addition to t o t a l cost is made by some sp e c i f i c addition to output, and at what price, when output has been thus augmented, i t s whole output can be sold. The fact that i t i s s e l l i n g in a 'perfectly competitive' market i s not announced to the firm by some independent or public agency, but must be established by the firm for i t s e l f (Shackle, 1972, p. 260) This information i s only garnered through a he u r i s t i c 279 process, and i s not contained e x p l i c i t l y or i m p l i c i t l y in the price system i t s e l f . Thus in response ' to the f i r s t issue, i t can be seen that conditions of supply and demand in the labour market are i p s i s f a c t i s conditions of general equilibrium. Even i f they could be regarded in p a r t i a l equilibrium terms however, this could not be enough to allow the price system to be the determining influence on market behaviour. Whether the terms of reference are general, or p a r t i a l equilibrium, therefore, the Keynesian c r i t i q u e of micro—economic theory holds. In regard to the second issue mentioned, the extent to which i t i s posible to talk of e f f e c t i v e demand other than in the context of demand for labour, both Pasinetti and Shackle make i t clear that one can. Pasinetti in his discussion, quoted e a r l i e r , of d i f f e r e n t sectors having d i f f e r e n t levels of u t i l i s a t i o n — a n d and therefore experiencing d i f f e r e n t levels of e f f e c t i v e demand—and Shackle in his discussion of the causes of lack of e f f e c t i v e demand being one of differences not so much in size as in composition of the product wage demanded and offered. The causes of the differences relate to the conditions of production of consumer goods, which require c a p i t a l investment. If c a p i t a l goods were homogeneous and malleable, the problem of investor uncertainty would not a r i s e , and there would be no problem of lack of e f f e c t i v e demand. If the c a p i t a l stock i s heterogeneous and largely non—malleable, as we know that that i t i s , then the processes of production associated with set of equipment w i l l be d i f f e r e n t , and so also w i l l be the conditions of investment in 280 each process. Since wages are paid in cash, and not in kind, reduction of output and therefore of income in any one sector w i l l represent a reduction of purchasing power throughout the whole economy. This manifests i t s e l f both in the drop in sales of consumer goods, and in the drop in sales of c a p i t a l goods to that sector. If a sector is operating below f u l l capacity u t i l i s a t i o n , that may represent a deficiency in e f f e c t i v e demand for that product, or an overestimation of investment p o t e n t i a l . S i m i l a r i l y a drop in output from f u l l capacity u t i i s a t i o n may be interpreted in either way. However that drop w i l l represent a decline in actual e f f e c t i v e demand in the economy as a whole, a decline which w i l l manifest i t s e l f more or less severely in any given sector depending on the conditions of investment in that sector. The conditions (borrowing short, lending long) moreover in which investment in housing is c a r r i e d out renders i t most susceptible to the effects of a deficiency in aggregate e f f e c t i v e demand. Thus even i f most sectors are close to f u l l capacity u t i l i s a t i o n , the production of housing i s less l i k e l y to be so. Looking a the problem in another way, l e t us assume that we can treat the patterns of investment in every sector as consisting of induced investment necessary to maintain current lev e l s of output, and autonomous investment necessary to bring the capacity of each sector up to the l e v e l of i t s maximum potential e f f e c t i v e demand. However though we may then sum a l l induced investment expenditures to give the l e v e l of aggregate 281 induced investment, t h i s i s not so easy with autonomous investment. This i s because in dealing with induced investment in t h i s situation we are dealing with a s t a t i c picture. So long as we avoid problems of double entry therefore, the amount of investment in any one sector can be added to the amount of investment in the next, without causing any changes in the evaluation of the investment in each sector when considered separately. This i s equivalent to saying that in the short run, the s t a t i c situation l e v e l s of investment are what they are, do not a f f e c t , and' therefore are independent of levels of investment in other sectors. But t h i s independence i s a l o g i c a l construct, only possible in a situation where nothing changes, one therefore in which there i s no causality. As soon as we introduce autonomous investment, t h i s s t a t i c picture changes. The p r o f i t a b i l i t y of autonomous investment can then only be assessed by reference to a l l other possible projects and the ruli n g rate of interest, and so autonomous investment in any sector cannot be treated independently of autonomous investment in any other sector. If the most p r o f i t a b l e projects are carried out f i r s t , t h i s w i l l more than l i k e l y enhance the p r o f i t a b i l i t y of their sectors generating a sit u a t i o n of cumulative causation. Again housing i s l i a b l e to be at a disadvantage in t h i s s i t u a t i o n . Even within the construction industry, investment yi e l d s a product which i s far more f l e x i b l e in i t s use than a home, namely o f f i c e construction. However housing, l i k e food and clothing, i s a basic 282 requirement for the reproduction of the labour force. It is therefore an important component of the wage good. But given the fact that control of investment i s in the hands of the entrepreneur, who has quite l i t e r a l l y better things to with his money (as he sees i t ) than invest in housing, then there is always l i k e l y to be a deficiency in the supply of housing. This l a s t argument must not be taken too far however. Pasinetti (1974), discussing the significance of the Cambridge savings, lack of e f f e c t i v e demand, and therefore Keynesian unemployment) writes that It shows that the necessity of maintaining f u l l employment keeps savings to being no more than a certain proportion of national income which in turn r e q u i r e s — i f a constant fraction of p r o f i t s i s saved—that the wage rate rises pari passu with productivity. The worst of Malthus' and Ricardo's fears, and the most frightening of Marx's predictions (that real wage rates would be nailed down to subsistence and a l l surplus above subsistence would be appropriated by the c a p i t a l i s t s ) thereby turn out to be incompatible with the maintenance of f u l l employment. In terms of exploitation one may say that exploitation i s indeed possible but only p a r t i a l l y — t o the extent allowed for by c a p i t a l i s t s ' consumption and accumulation. Once these are provided for, f u l l employment can be maintained only i f the whole increasing productivity goes to wages. And yet there is very l i t t l e ground for complacency, even i f the outlook i s no longer so bleak as i t used to be among the c l a s s i c a l representatives of the 'dismal science'. If the so much feared increased immiseration of the working class i s avoided, and a continually growing l e v e l of per capita wages takes place instead, one can hardly attribute i t to any i n t r i n s i c merit of a c a p i t a l i s t system. What i s taking place is imposed by the requirements of Equation (in the form which y i e l d s i t shows that which would mean excess 283 su r v i v a l . The system could not last o t h e r w i s e — i t would be doomed to slump and collapse. (Pasinetti, 1974, p. 102) Thus although entrepreneurs may not want to be committed to provide housing and w i l l be cautious about doing so, nonetheless some degree of accommodation w i l l be provided. Discussion of the Cambridge Equation brings us to the question of the operation of e f f e c t i v e demand in the long run. We have seen that e f f e c t i v e demand can vary for di f f e r e n t ectors' outputs. Using the Cambridge equation as a basis, we may investigate the p o s s i b i l i t y that e f f e c t i v e demand w i l l also vary as between income and soc i a l groups. 284 8.5 E f f e c t i v e Demand And The Provision Of Housing: The Long Run We return, as promised, to the Cambridge equation, discussed e a r l i e r . There i t was merely used to derive a statement about the l o g i c a l course of urban growth over time. Here we wish to use i t to show what the implications are i f a number of groups of income earners are considered. Pasine t t i ' s analysis begins with the proposition that aggregate savings can be s p l i t into a number of categories S = sw(vl + P„) + s cP c H = U +P I = S P , P c are the p r o f i t s earned on workers' and and c a p i t a l i s t s ' incomes respectively, and a l l other terms as before. Let P w + L Kw where K w is the share in the c a p i t a l stock that workers' loans to c a p i t a l i s t s ' buy them. Then K" st-sw' K 5 t -s w" K K K " 8 I st-sw I st-sw thus the.rate of p r o f i t i s ?_ _ l I s w V • /SwSc _K. _ Su K_\ K " ^ -s w K s t-s w K V S c - S w ' l s t-s w Y/ and the share of p r o f i t s in income i s 2 8 5 \ ~ S t - S w Y Sc-5w \SC-Sw I S t - S w * \ / If we assume that L equals the rate of p r o f i t , these two relations reduce to 2 - = J - . i = __P_ _ i . i . - i i . J L - J L a A -v ~ S c ' y ~ S.'K " Y ~ «c J * 1 Y thus in the long run, workers' propensity to save, though influencing the d i s t r i b u t i o n of income between c a p i t a l i s t s and workers does not affect the d i s t r i b u t i o n of income between p r o f i t s and wages, nor does i t have any influence on the rate of p r o f i t . These results derive from the i n s t i t u t i o n a l p r i n c i p l e that wages are proportional to output, and that p r o f i t s are proportional to c a p i t a l owned. The implications of th i s are that K Kw |<w K K ~ Kc which implies that I L = L i . e . , p r o f i t s are distributioned in proportion to savings contributed. At th i s point, the par t i c u l a r types of incomes out of which savings are made become relevant. ?w = Pc = 1 S«(W+P W> IcZ ^ Savings out of wages always turn out be equal to workers' extra consumption out of p r o f i t s (extra consumption meaning in excess of what the c a p i t a l i s t s would have consumed i f these p r o f i t s had remained to 286 them—[because the c a p i t a l i s t s would have them]—). P A The fact that -~- = — also establishes the fact that i t is the c a p i t a l i s t s ' propensities to save that determines "the ra t i o of p r o f i t s to savings for a l l savings groups and consequently also the income d i s t r i b u t i o n between p r o f i t s and wages and the rate of p r o f i t for the whole system." If we generalise the analysis to include a rate of interest p d i f f e r e n t from the rate of p r o f i t equal to — a , the results are K w S Sc s modified, but not altered. Returning to the rel a t ion -rr- ~ -rr * -rf« ^ i t i s clear that t TC does not affect t h i s . Since in equilibrium 1=3 and ~ - 9*\ ' i f c follows that Q = f^- - S ' , so that This equation represents in fact a more general version of the 'Cambridge equation'. It shows that the natural rate of growth and the c a p i t a l i s t s ' propensity to save determine the rate of p r o f i t on c a p i t a l i s t s ' c a p i t a l f i r s t of a l l , independently of anything else, and therefore independently of the rate of interest. (Pasinetti,1974) Let Thus we have two rates of return to c a p i t a l : a rate of p r o f i t on c a p i t a l i s t s ' c a p i t a l and a rate of interest on workers' c a p i t a l lent to the c a p i t a l i s t s . The o v e r a l l rate of p r o f i t needs to be s l i g h t l y redefined as K Ku+K* Kc K K w K Kc K K 2 8 7 J5w - JmA&..X- — ——— •, as before, and by straightforward K S0- Su I Sc-Sw algebraic manipulation can be shown to be S u 73*^ ) , where K. i s the capital—output r a t i o corresponding to yt - (Vjc)^ n. K S " X ^ ~ 5 w w ° I Then K _fLq„ where (-Sc - 3 W ^ ^ v ^ X S^" 1 9 ^ * ( S c - |*.Sw) + ScSw(l-^ * when either - 1 , so that c = 7T , or S w = 0 , then % = 1 , and we return to the results obtained before that of IT = A s w e noted e a r l i e r H^c )> Sc. so that the greater the difference between the rate of p r o f i t and the rate of interest, the greater the amount of investment in any period as one would expect on Keynesian p r i n c i p l e s . We may introduce many groups of savers also. There may be both groups of entrepreneurs with d i f f e r e n t savings propensities S c > 5»c > Sc,>-• •> , and groups of workers S w > S w > S w > • • • >SU >/0. - In the case of c a p i t a l i s t s , the t h r i f t i e s t group, that i s S, w i l l have a higher growth rate than the others, and, growth being exponential, w i l l eventually come to dominate the rest. 288 Therefore only their savings propensity is relevant to the determination of the rate of p r o f i t . In the case of the workers, " the eff e c t s of di f f e r e n t propensities to save i s that of determining a d i f f e r e n t equilibrium share into the ownership of the t o t a l c a p i t a l stock for each of the workers' groups. A l l groups of workers... w i l l therefore coexist on the equilibrium growth path." For P a s i n e t t i , the important implication i s that therefore the weighted average of them propensities to save, Sj* , need only to be substituted into the equation for ^ to obtain the general rate of interest d i f f e r e n t from the rate of p r o f i t , and many groups of savers. However other implications may be drawn, e.g., rate of growth determines income d i s t r i b u t i o n . The effect of this analysis i s to suggest that income receivers may be grouped in terms of their 'asset preferences' (Kregel, 1980). This links this analysis with Minsky's analysis of c a p i t a l i s t dynamics in terms of p o r t f o l i o preferences, and Shackles preferred composition of the product wage. Grouping income receivers in thi s way therefore suggests that e f f e c t i v e demand w i l l vary across these groups. Kregel (1980) implies that t h i s i s so. He writes Y = W + (±- r)£ - W + ^ Y = rP 3 = I = sV +. s(w)P + T ? P~= 1+ (l-*)D+ sW 289 Here Y^ i s firms' incomes, Y^ ^ s household incomes, J) i s dividends received by households, and Y" is the proportion of p r o f i t s firms retain for internal funding of investment. Comparison of Kregel's and Pasinetti's investment equations shows that the two approaches are in p r i n c i p l e compatible. Pasinetti writes Kregel writes 1, s"W+ s(i~r)P + rP = s(y+u-Y)p) + Yp and /. (i-r)P2p w , rP s scP , r = s c an<*. SEE S w with Kregel's terminology on the l e f t hand side of each def i n i t i o n . Kregel then manipulates his d e f i n i t i o n s to obtain two alter n a t i v e d e f i n i t i o n s of the rate of p r o f i t ? J-sV _ I/K- s(W/K) = 9 - sCVTlK) These formulae for TC obtain in the situation where there i s only one s o c i a l class, which owns shares in the firms from which they receive wages and dividened received from shares owned in the firms. A l l households in t h i s class have the same savings propensity . (These assumptions are designed to give as much as possible to neo-classical theory, as both he and Pasinetti argue the existence of functional income classes do not affect in any 290 way the macro—relation Jf ^ If §t\ )• I n r e 9 a r Q l to the f i r s t variant of t h i s formula, Kregel writes that the expression " S ( l — 0 + r i s just another way of expressing the propensity to save out of t o t a l p r o f i t s (distributed + undistributed)". The second version of the formula brings out this point more c l e a r l y since r+ d i r e c t l y equivalent to Pasinetti's S c . It also takes into account market power relations in the expression The firms in t h i s formulation subsume the capacity of the c a p i t a l i s t s for the combination of their investment decisions and the funding of t h i s investment... determines the output of consumption commodities and the nominal value of household income... available to purchase them. If a l l households are in an equal position, (equal"W, s,T> ), they a l l have equivalently lower consumption when the firms as a whole carry out a higher rate of investment [though the nominal value of their shareholdings might r i s e ] . . . But there i s no need to l i m i t the analysis to a single household saving propensity and therefore received income. The l i m i t i s a d i f f e r e n t savings propensity for each l e v e l of income. But here exactly the same mechanism works. Those households with a low income consume a higher proportion of their income [this i s i m p l i c i t in the d e f i n i t i o n of the consumption function] and therefore have a lower wealth position an d a lower a b i l i t y to save. In such conditions a higher rate of investment implies a proportionately greater r i s e in income for high income classes and a proportionately greater cut in real consumption for low income households. The 6-1 balance i s achieved by reducing the consumption of low incomes through r i s i n g prices that high households can more ea s i l y absorb. It is not d i f f e r e n t amounts of savings or propensities to save but the a b i l i t y to save that a given income size allows which determines who gives up consumption to allow the increased investment. On the tack of r e a l i t y , i t should be noted that t h i s applies regardless of types of income recieved (as landlords in the nineteenth and rentiers in the twentieth century w i l l t e s t i f y ) . 291 Thus the implication that the brunt of economic growth and accumulation w i l l be borne by low incomes irrespective of source. As long as real purchasing power i s transferred from those who consume a higher proportion of their income to those who consume a lower proportion, no conceivable s o c i a l scheme w i l l bring equality in the burden of growth and investment. A dynamic growing system that allocates output v i a the 'price mechanism' w i l l produce income inequality. (Kregel, 1981) As we have said, the main use of the Cambridge equation i s to establish propositions about the relations necessary to maintain stable long period growth. However t h i s need not prevent us from asking what w i l l happen i f ever these relations are not maintained. The obvious result is that this f a l l w i l l be proportionately higher for low income groups. Moreover those sectors which provide for low income groups w i l l suffer the greatest lack of e f f e c t i v e demand of a l l . Workers in these sectors w i l l suffer from low incomes as a re s u l t . Once again we see that the price system i s i n e f f e c t i v e to deal with t h i s problem. Therefore as Robinson says the problem of unemployment being soluble in p r i n c i p l e by in creasing the volume of investment, the question as to the content of that investment becomes c r u c i a l (Robinson, 1971). Simple macro—economic f i s c a l incentives to stimulate aggregate e f f e c t i v e demand via consumer demand management w i l l therefore be of only limited effectiveness. Only in conditions such as the Depression w i l l those measures be of any real effectiveness. It i s almost redundant to make the point that low income housing i s a prime example of a sector l i k e l y to suffer from a 292 lack of e f f e c t i v e demand, in spite of growth in the economy as a whole, and indeed in conditions of general i n f l a t i o n . If there were no variation in e f f e c t i v e demand between sectors, i n f l a t i o n would only result from aggregate e f f e c t i v e demand increasing beyond the point of f u l l capacity output. In the case of housing in general, and low income housing in p a r t i c u l a r , the conditions of investment are such that f u l l capacity output in this sector is l i k e l y to be reached only after f u l l capacity output has been reached in almost a l l other sectors. If aggregate e f f e c t i v e demand i s simply pushed up to the point where t h i s lack of ef f e c t i v e demand in the housing sector is overcome, the res u l t i n g i n f l a t i o n a r i s i n g in other sectors w i l l be s u f f i c i e n t to frustrate the policy objective of increasing e f f e c t i v e demand in t h i s sector. As a conclusion to th i s exposition of post-Keynesian approaches to the theory of urban growth therefore, we w i l l provide an analysis of housing supply from this perspective. As in the case of mortgage finance i n s t i t u t i o n s , housing market behaviour which appears anomalous from the neo-classical perspective w i l l appear exactly as expected from the standpoint of post-Keynesian theory. Before concluding this section however le t us once more review the analogies between the housing market and the labour market which permits the application of Keynesian economic concepts to an analysis of t h i s sector. According to marginal productivity theory, an employer w i l l take on labour (have u n f i l l e d job vacancies) up to the point 293 where the marginal product of the la s t man employed equals the cost of employing him (of supplying him with a job). If the marginal productivity of labour f a l l s below the marginal cost of employing labour, then the cost of providing jobs for labour w i l l be greater than the return labour provides for the employer. The neo-classical solution is that labour must therefore cut i t s wage in order to reduce i t s cost below the marginal product i f the l e v e l of vacancies is to be increased. The Keynesian c r i t i q u e begins here. The same language can be used to describe the rental housing market. The landlord w i l l supply vacant apartments so long as rent received from the la s t tenant is at least as great as the marginal cost of doing so. If the rent received from the tenants f a l l s below the marginal cost of supplying apartments, there w i l l be a reduction in the supply unless tenants agree to a rent increase. An increase in rents in the housing market i s in p r i n c i p l e the same as a wage cut in the labour market, since both involve a transfer of income from one economic class to another. These l a s t two sections have been devoted to proving both that this analogy is v a l i d , and that the Keynesian c r i t i q u e of neo-classical theories of unemployment is equally v a l i d when applied to the neo-classical account of housing provision, i . e . , when i t i s not applied in macro terms, but to pa r t i c u l a r sectors, and to par t i c u l a r s o c i a l groups. As an added bonus i t also provides a c r i t i q u e of the neo-classical c r i t i q u e of rent 294 control, as well as an account of the behaviour of the f i n a n c i a l variables involved in the economic system, i . e . , mortgage f inance. However, as Minsky puts i t , Keynesian economics provides a f i n a n c i a l theory of fluctuations in real investment, and an investment theory of fluctuations in real demand. The investment theory of fluctuations in real demand having been thoroughly examined, we now turn to an account of the f i n a n c i a l theory of fluctuations in real investment for housing. 8.6 Housing Supply In A Keynesian Perspective Up t i l l now this account of the operation of the housing market has involved a long c r i t i q u e of existing theory in order to make clear i t s advantages over the previous one, and to rescue the occasional salient observation from the morass of inapplicable theory surrounding i t . Ha