UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The contribution of non-renewable natural resources to economic development : the case of copper in Zambia Panaĭotov, Todor 1978

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1978_A1 P35.pdf [ 16.56MB ]
Metadata
JSON: 831-1.0094542.json
JSON-LD: 831-1.0094542-ld.json
RDF/XML (Pretty): 831-1.0094542-rdf.xml
RDF/JSON: 831-1.0094542-rdf.json
Turtle: 831-1.0094542-turtle.txt
N-Triples: 831-1.0094542-rdf-ntriples.txt
Original Record: 831-1.0094542-source.json
Full Text
831-1.0094542-fulltext.txt
Citation
831-1.0094542.ris

Full Text

THE CONTRIBUTION OF NON-RENEWABLE NATURAL RESOURCES TO ECONOMIC DEVELOPMENT: THE CASE OF COPPER IN ZAMBIA by THEODORE PANAYOTOU B.A. Un ivers i ty of Athens, 1972 M.A. York Un i ve r s i t y , 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept th i s thes i s as conforming to the THE UNIVERSITY OF BRITISH COLUMBIA in required standard February, 1978 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thou t my w r i t t e n p e r m i s s i o n . Department o f ECOh/OM <C f The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 Date T £ B> RJD Aft-V y22 ^IR i • THE CONTRIBUTION OF NON-RENEWABLE NATURAL RESOURCES TO ECONOMIC DEVELOPMENT: THE CASE OF COPPER IN ZAMBIA Research Supervisors: Professors Anthony D. Scott and Gideon Rosenbluth ABSTRACT This thes i s focuses on three aspects o f long-run planning fo r mineral-dependent economies: ( i ) modell ing mineral deplet ion fo r optimal cap i ta l accumulation; ( i i ) s p e c i f i c a t i o n and est imation of a mining cost funct ion and a non-mining production func t i on ; and ( i i i ) measurement of cap i t a l and natural resource stocks. Zambia, a country t o t a l l y depend-ent on copper, i s used as a case study. The planning model integrates the Ramsey cap i t a l model with the Hotel l i n g - S co t t exhaust ib le resource model, u t i l i z i n g recent advances in dynamic opt imizat ion and dua l i t y theory. The optimal rate of resource ex t rac t ion requires that marginal va r i ab le cost plus user cost be equal to the world p r i ce of the minera l . The user cost of the resource grows at the rate of i n te re s t plus the rate of populat ion growth minus the e f f e c t of deplet ion on ex t rac t ion cost . The optimal a l l o c a t i o n of resource revenues equates the marginal benef i t s from a l t e r n a t i v e uses: ( i ) current consumption to r a i se the standard of l i v i n g of a growing populat ion; ( i i ) mining investment to maintain production in the face of de te r i o ra t ing resource q u a l i t y ; and ( i i i ) non-mining investment to create an i ndu s t r i a l base as an a l t e r n a t i v e to the deplet ing resource. A var i ab le cost funct ion (VCF) f o r the mining industry is derived from the assumption that the mining f i rm minimizes the var i ab le cost of producing a given output. The parameters of a t rans log VCF f o r Zambian copper mining are estimated and found to be cons i s tent with the under-i i l y i ng theory. It i s a l so found that domestic labour i s only imperfect ly subs t i tu tab le f o r imported machinery and that labour demand i s not very responsive to changing wages. A substant ia l port ion of the empir ica l cont r ibut ion i s the con-s t ruc t i on of comprehensive data ser ies for Zambia using the perpetual inventory approach to cap i t a l s tocks, D i v i s i a , Indexes, and the neo-c l a s s i c a l theory of renta l p r i c e s . Production f igures f o r i nd iv idua l mines are used to construct aggregate cumulative ser ies to which a curve of d iminishing increments i s f i t t e d to obtain andestimate of the Zambian copper resource. i i i TABLE OF CONTENTS Page ABSTRACT . i TABLE OF CONTENTS. . . . . . . . . . . . i i i LIST OF TABLES . . v i i LIST OF FIGURES .xi ACKNOWLEDGEMENTS xi i KEY NOTATION .x iv CHAPTER 1 INTRODUCTION 1 1.1 The Problem . . . . 1 1.2. Previous Studies 5 1.3. Scope and Out l ine of the Present Study 7 Footnotes 12 2 COPPER MINING AND ZAMBIAN ECONOMIC GROWTH: THE NEED FOR LONG TERM RESOURCE AND DEVELOPMENT PLANNING....14 2.1. General Background 14 2.2. The Zambian Copper Industry, 1920-1964 15 2.3. Economic Condit ions in the Non-mining Sector, 1920-1964 21 2.4. Development Planning and Mineral Resource Management in Zambia, 1964-1974 25 2.5. The Need fo r a Long-run Development P l an . . 36 Footnotes 40 i v Page CHAPTER 3 MODELLING OPTIMAL MINERAL DEPLETION IN DEVELOPING ECONOMIES: THE ROLE OF PRODUCTION TECHNOLOGIES 45 3.1. Assumptions 46 3.2. Formulation of the Planning Problem 49 3.3. Object ive Function and Terminal Conditions 52 3.4. I n i t i a l Resource Condit ions and Dynamic Constra ints 55 3.5. Technological Constra ints : Mining and Non-mining Producti on Technol ogi es 57 3.6. The Complete Model and Opt imal i ty Condit ions: An Economic In terpreta t ion . .64 3.7. The Role of Production Technologies 74 Footnotes 82 4 MINING AND NON-MINING PRODUCTION TECHNOLOGIES: FUNCTIONAL SPECIFICATION 86 4.1. Mining Sector: A Translog Var iab le Cost Function 86 4.2. Non-Mining Sector: A Translog Production Func t i on . . . .TOO 4.3. Further Theoret ica l Concepts: Subst i tut ion and Pr i ce E l a s t i c i t i e s 104 Footnotes 109 5 DATA: REQUIREMENTS SOURCES AND METHODOLOGY 112 5.1. Construct ion of Capi ta l Stock Ser ies : The Perpetual Inventory Approach 113 5.2. Construct ion of Rental P r i ce s : The Jorgenson-G r i l iches-Diewert Procedure 119 5.3. Aggregation of Non-mining Inputs and Outputs: The D i v i s i a Index.... 129 5.4. Mining Inputs and Outputs: Aggregation of Imported Var iab le Inputs 133 V Page CHAPTER 5 5.5. Est imation of the Copper Resource Stock 139 Footnotes 162 6 STOCHASTIC SPECIFICATION AND METHOD OF PARAMETER ESTIMATION 1 70 6.1. Est imating Equations and Stochast ic S p e c i f i c a t i o n . . . . . 1 7 0 6.2. Est imation Technique and Computational Algorithm 173 6.3. Problems of Autocorre la t ion and the L ike l ihood Ratio Test 174 Footnotes 177 7 EMPIRICAL RESULTS AND THEIR INTERPRETATION 178 7.1. Estimated Mining Technology 178 7.2. Estimated Non-mining Technology 204 Footnotes 211 8 TOWARDS A NUMERICAL SOLUTION 214 8.1. The Socia l Welfare Function 215 8.2. Long-run Pr ice Project ions 220 8.3. A Discrete Time Approximation to the Planning Model...226 8.4. So lut ion Algorithms 231 Footnotes 238 9 SUMMARY AND CONCLUDING REMARKS 242 BIBLIOGRAPHY 249 STATISTICAL SOURCES 259 APPENDIX A TECHNICAL ASPECTS OF MINING AND ORE TREATMENT .260 vi Page APPENDIX B RELAXING SOME ASSUMPTIONS OF CHAPTER 3 264 APPENDIX C SUPPLEMENTARY DATA TABLES 265 vi i LIST OF TABLES TABEE Page I Prospecting Licences Zambia, 1922-1930 . . . . . . . . . 2 0 II Zambian Copper Mines: Discovery and I n i t i a l Capacity 20 III Copper P r i c e s , Growth Rates, Mining Share in GDP, Exports and Employment, Zambia 1954-1964... 23 IV Development Planning in Zambia: Performance of TDP and FNDP. 32 V Estimated Capita l Stocks f o r Non-mining Sectors , Zambia 1945-1973 (Kmn 1967 p r i c e s ) . . . . . . . . . .116 VI Estimated Stocks o f Fixed and Var iab le Capi ta l in Mining and Corresponding Flows o f Replacement and Net Investment Zambia 1945-1970 (Kmn 1967 pr i ces ) 120 VII Computed Rental Pr ices f o r Non-mining Capi ta l Stocks Zambia 1954-1972 .. . .126 VIII Computed Rental Pr ices (for Mining Var iab le Capi ta l (Machinery and Equipment), Zambia 1945-1970 128 IX D i v i s i a Quantity and Imp l i c i t P r i ce Indices fo r Non-mining Output, Capita l and Labor, Zambia 1954-1972 (1967 Base Year) 134 X Copper Production and Copper P r i c e s , Zambia 1924-1975 (Output: Metr ic Tons Pr ices Kmn Per Metr ic Ton) 136 XI Mining Employment of Af r i cans and Europeans and The i r Respective Annual Earnings, Zambia 1945-1970 (Employment in '000 o f Persons, Earnings in '000) 138 XII D i v i s i a Pr i ce Index o f Imported Var iab le Input and Corresponding Imp l i c i t Quantity Index, Total Var iab le Cost, Unit Cost and Unit Quant i t ies o f Var iab le Inputs i n Mining Zambia 1945-1970 140 XIII Average Grade (a) of Ore Mined in Three Zambian Mines 1932-1975.. . . . . 153 XIV Estimated Ore Reserves, Average Grade, Copper Reserves and Recoverable Copper Reserves, Zambia 1931, 1945-1975 (Reserves: '000 Metric/Tons Grade: Copper Content Per Ton of Ore 156 vi i i TABLE Page XV Ore Product ion, Copper Product ion, Cumulative Ore Production and Cumulative Copper Product ion, Zambia 1944-1975 ('000 of Metr ic Tons) 157 XVI Cumulative Ore Tonnage and Recoverable Cumulative Copper tonnage-Zambia 1945-1975 ('000 Metr ic Tons) 159 XVII Est imation o f "fhe Ult imate Copper Resource of Zambia... 160 XVIII Copper Resource Stock, Ore Resource Stock and Corresponding Average Grades Zambia 1945-1975 Stocks: '000 Metr ic Tons. Grades: Copper Content per Ton of Ore ...162 XIX Mining Var iab le Cost Function (Models A & B): L ike l ihood Ratio Test S t a t i s t i c s and C r i t i c a l x 2 Values . . . . .182 XX Mining Var iab le Cost Function (Model A): Parameter Estimates 183 XXI Mining Var iab le Cost Function (Model A) : Predicted Shares, AES, and Pr ice E l a s t i c i t i e s f o r Selected Years 185 XXII Mining Var iab le Cost Function (Model B): Parameter Estimates . . . .192 XXIII Mining Var iab le Cost Function (Model B): Shares, Shadow P r i c e s , and E l a s t i c i t i e s f o r Selected Years 194 XXIV Mining Var iab le Cost.Funct ion (Model C): Parameter Estimates 198 XXV Mining Var iab le Cost Function (Model C): A l l en E l a s t i c i t i e s o f Subs t i tu t ion (a) and Pr ice E l a s t i c i t i e s (n).199 XXVI Mining Var iab le Cost Function (Model C): Inverse E l a s t i c i t i e s o f Subs t i tu t ion and of Intens i ty 200 XXVII Mining Var iab le Cost Function (Model C): Predicted Shares and Inverse S t a t i c Shadow Pr ices 202 XXVIII Non-Mining Production Function Parameter Estimates 206 XXIX Non-Mining Production Function (F.D.A.) : Predicted Shares and Marginal Products 208 ix TABLE Page XXX Non-mining Production Function (F.IO.A.) A l l en E l a s t i c i t i e s o f Subs t i tu t ion and Pr ice E l a s t i c i t i e s ..209 XXXI I t e ra t i ve Procedures f o r Numerical So lut ion 231 XXXII Gross Fixed Investment.in the Non-mining Sectors , Zambia 1945-1972 (Kmn 1967 pr i ces ) 267 XXXIII Gross Fixed Investment Def lators f o r the Non-mining Sectors , Zambia 1954-1972 268 XXXIV Mining Sector: Total Gross Investment in Machinery, Zambia 1945-1970 . . . . .269 XXXV Mining Investment Def lators f o r S t ructures , Machinery and "their Aggregate, Zambia 1945-1970 (1967=1.00) 27* XXXVI Capi ta l Consumption Allowances fo r the Non-minfng Sectors , Zambia 1954-1972 272; XXXVII Der ivat ion o f D i rect Income Tax Rate Zambia 1954-1972 273 XXXVIII Mining: Capita l Consumption Allowances and Exogenous Rate o f Return, Zambia 1946-1970 (Allowances in Kmn) 274 XXXIX Mining: Gross Domestic Product, Ind i rect Taxes Rental B i l l and Corporate P r o f i t s , Zambia 1945-1970 (Kmn)...276 XL D i rec t Taxes, Net National Product at Factor Cost and Tax Rates Zambia 1945-1970 278. XLI Non-mining Gross Domestic Product at Factor Cost by Sector of O r i g i n , Zambia 1954-1972 (Kmn) 27.9 XLII Gross Domestic Product Def lators by Industr ia l Sector , Zambia 1954-1972 (1967=100). £80 XLI11 A f r i c an and European Employment in the Five Non-mining Sectors , Zambia 1954-1972 (Number o f Persons) 2:81 XLIV Average Annual Earnings of Af r icans and Europeans Employed in the Five Non-mining Sect ions , Zambia 1954-1972 (Kwachas) .283-XLV Mining Employment and Average Earningsoof PEuropean-arid A f r i can Mining Labor, Zambia 1945-1953 285 X TABLE Page XLVI Ore Reserves, Average Grade of Reserves, and Ore Hoisted by Indiv idual Copper Mines, Zambia 1931 and 1931 ,-1975 (Ore: M i l l i on s of Metr ic Tons; Grade in Copper Content Per Ton of Ore) .286 xi LIST OF FIGURES FIGURE Page 1 Relat ion Between Cumulative Ore Tonnage and Cumulative Copper Tonnage: The "Curve of Diminishing Increments" 147 2 Map of the Copperbelt and Location of Major Deposits 152 3 Typica l Flowsheet of Operations 263 xi i ACKNOWLEDGEMENTS In the preparation of this study I have benefitted greatly from the suggestions of many people. Special acknowledgement and thanks should go to my dissertation committee consisting of Professors P.G. Bradley, H.F. Campbell, G.B. Hainsworth, G. Rosenbluth, and A.D. Scott. Professor Scott as Chairman during the earlier stages provided initial encouragement for the project and made a number of most important comments. Moreover the influence of his teachings is more extensive than explicitly stated. Professor Rosenbluth, as Chairman during the difficult last six months provided me with invaluable guidance and constructive criticism. Professor Hainsworth provided helpful advice on the development aspects and made a number of valuable suggestions for improving exposition. Professor Campbell offered guidance and encouragement in the earlier phases of this project and Professor Bradley made particularly helpful comments on the mining aspects of the study. My intellectual debts in writing this thesis are evident from the foot-notes, but Professors Diewert, Woodland, and Helliwell deserve special acknowledgement. I am particularly indebted to Professor Diewert who in addition to valuable suggestions, provided me with the proofs appearing in section 4.1 of the thesis. I have also been extremely fortunate in receiving a great deal of advice and assistance from my colleaguesMMohammed Khaled and Chris Clark, to whom I am deeply grateful. I would also like to acknowledge the valuable comments of my colleagues Thomas Burlington, Donal Donovan, Brenda Lundman, Edward Morey and of the participants of Ph.D. Thesis Workshop at the University of British Columbia. xi i i Malcolm McPherson, graduate student at Harvard Un ivers i ty and consultant to the World Bank, was kind enough to provide me with a sub-s t a n t i a l volume of unpublished data and he lpfu l adv ice. His help is g r a t e f u l l y acknowledged. I a lso wish to thank Professor Herbert Drechsler of the Faculty o f Commerce and Business Adminis trat ion at the Un ivers i ty of B r i t i s h Columbia, S. Cunningham of the Sec re ta r i a t of the United Nations, and Robert Sargent, Vice President of American Metal Climax, f o r t h e i r help in obta in ing mining data. The completion of th i s thes i s was a long and arduous task. Much of the c r e d i t should go to my s i s t e r - i n - l a w , Androula Panayiotou, who d i l i g e n t l y typed success ive dra f t s of a cumbersome manuscript and remained cheerful despite the obs tac les . I would a lso l i k e to express my thanks to Nasim Damji, June Janson, and May McKee fo r an exce l len t job in typing the f i n a l d ra f t and to Mary Louise Evans fo r proof-reading and other ass i s tance. This thes i s a l so benef i t ted from f i nanc i a l support through the Programme in Natural Resource Economics at U.B.C. which i s funded by Canada Counc i l . F i n a l l y , I express my spec ia l grat i tude to my parents f o r provid ing continual encouragement and support throughout my educat ion. This thes i s i s dedicated to them. T. Panayotou Vancouver, B.C. January, 1978. xi v KEY NOTATION (unless otherwise ind icated) a: parameters o f the mining var i ab le cost funct ion b: parameters o f the non-mining production funct ion c: un i t va r i ab le cost in mining g: renta l p r i ce of imported mining inputs h: non-mining labor per cap i ta k: non-mining cap i ta l per cap i ta a: mining labor per unit of output m: cost -minimiz ing quant i t ie s of imported inputs (machinery plus expatr ia te labor) per uni t o f output n: growth rate of labor force and population p: world metal (copper) p r i ce q: copper output per cap i ta r: mineral resource stock per cap i ta s: non-mining investment per cap i ta t : time u: u t i l i t y ( funct ion) v: mining' investment per cap i ta w: mining wage rate x: per cap i ta consumption y : per cap i t a quant i ty of the homogeneous commodity Y z: mining s t ructures per cap i ta E: populat ion G: average grade of cumulative production H: Hamiltonian XV Q: mineral (copper) output R: the mineral (copper) resource stock T: time horizon and terminal time U: u t i l i t y funct ion Y: a homogeneous commodity which can be e i t h e r consumed or invested Z: mining s t ructures 0: dynamic shadow pr i ce of the non-mining cap i ta y. dynamic shadow pr i ce of mining s t ructures 6: deprec iat ion rate(s ) e: e l a s t i c i t y o f marginal u t i l i t y with respect to consumption n: p r i ce e l a s t i c i t y o f parameter of the u t i l i t y funct ion g: non-subsistence labor force per cap i ta (a constant) p: discount rate a : e l a s t i c i t y o f s u b s t i t u t i o n , inverse subs t i tu t i on and i n ten s i t y T: trend var i ab le <j>: non-mining production funct ion in per cap i ta terms ty: dynamic shadow pr i ce of the mineral resource w: s t a t i c shadow pr ices CHAPTER 1 INTRODUCTION The problem of achieving economic development through e f f i c i e n t ex-p l o i t a t i o n of non-renewable natural resources has received very l i t t l e a t tent ion in comparison to the voluminous l i t e r a t u r e on development through a g r i c u l t u r a l exports , surplus l abor , fore ign a i d , and external borrowing. 1 However, the share of non-renewable resources in to ta l exports from the less developed countr ies (LDCs) has always been high and s tead i l y r i s i n g . Yet many mineral export ing LDCs, such as Zambia, L i b e r i a , Z a i r e , C h i l e , Peru, B o l i v i a , Jamaica, Burma, Botswana, Tha i l and, and others , enjoy no more than subsistence leve l s of consumption, and while t h e i r natural cap i ta l i s being depleted, they have done l i t t l e in the way of accumulating reproducib le c a p i t a l , to provide an a l t e r n a t i v e source of growth. The ob jec t i ve of the present study i s to construct a model of economic development through optimal exp lo i t a t i on of a non-renewable resource and i n -d icate how a numerical so lu t ion might be obtained. In th i s context we 3 develop a genera l ized var i ab le cost funct ion and estimate the Zambian mining and non-mining production technologies using data ser ies constructed as part of th i s . s tudy . Zambia was chosen as the c l a s s i c example of extreme depend-ence on a non-renewable resource in order to sharpen the r e s u l t i n g imp l i c a -t i ons . 1.1 The problem Economic development has been def ined as the process by which an eco-4 nomy in a s ta t ionary s tate a t ta ins a s i g n i f i c a n t sustained r a te .o f growth in 5 per cap i ta income as a per s i s ten t long-run feature. Thus, economic develop-ment i s d i s t ingu i shed "from such processes as sporadic growth sustained - 2 -g p r imar i l y by exogenous f o r c e s . " By d e f i n i t i o n , in minera l -export ing LDCs, growth i s exogenously gener-ated by the world mineral demand. The l a t t e r , however, cannot be r e l i e d upon to susta in growth as a per s i s ten t long-run feature of the economy. Short-run f l uc tua t ions a s ide , mineral pr ices may co l lapse under the pressure of uncontro l lab le factors such as the discovery of new sources of supply, the development of subs t i tu te s , or a change in technology and ta s tes .^ This coupled with the increas ing cost o f mining success ive ly i n f e r i o r ores , may lead to the exhaustion of the country ' s commerically exp lo i t ab le mineral de-o pos i t s . Although an export -or iented mineral industry cannot be r e l i e d up-on to generate sustained growth, i t can, in f a c t , provide the f i n a n c i a l means in terms of savings and fore ign exchange f o r the development of a l t e r n a t i v e sources of growth such as commercial a g r i cu l tu re and manufacturing. Many minera l -export ing count r ie s , however, have at ta ined no more.than "sporadic growth". Even the act ions of speculators at the London Metal Ex-change a re , sometimes, s u f f i c i e n t to turn an LDCs growth rate from po s i t i ve 9 to negative overnight. Th i s , o f course, i s a d i r e c t consequence of the dominating share of a s ing le commodity in a country ' s exports and nat ional income. This v u l n e r a b i l i t y of mineral -exporters (and indeed a l l primary-exporters) to world market developments points to the need f o r concerted act ion on t h e i r part to s t a b i l i z e mineral (or other primary commodity) p r i ce s . However, i f we exclude the Organizat ion of Petroleum Exporting Countries (OPEC) , 1 0 LDCs have been genera l ly unsuccessful in exerc i s ing mono-p o l i s t i c power to achieve high and s tab le p r i c e s . Unl ike OPEC, other pr im-ary exporters ' shares in the world market are r e l a t i v e l y small and the e l a s -t i c i t y o f non-LDC supply r e l a t i v e l y h igh, and unl ike o i l , most primary com-- 3 -modif ies face e l a s t i c world demand. 1 1 Nor has the a l t e r n a t i v e of c reat ing commodity buf fer stocks met with much success due to severe problems of f i n -12 ancing and managing such stocks. Moreover, even i f s tab le pr ices could be e s tab l i shed , LDCs could not r e l y on mineral exports alone f o r sustained long-run growth because of the f i x i t y and non-renewabi1ity of these resources. The v u l n e r a b i l i t y of mineral revenues to uncontro l lab le short-run p r i ce f l uc tua t ions and the e x h a u s t i b i l i t y o f mineral resources point to the need 13 fo r progress ive reduct ion of a country ' s dependence on any given minera l . This involves a long process o f s t ruc tu ra l change requ i r ing considerable i n -vestment in non-mining a c t i v i t i e s in order to create an i n d u s t r i a l base as an a l t e rna t i ve source of growth to the deplet ing mineral r e s o u r c e . 1 4 At the same time, increas ing amourjts of investment in mining s t ructures are required to maintain or increase mineral production in the face of d iminishing access-15 i b i l i t y and f a l l i n g grades of the mineral resource. Last but not l e a s t , increas ing amounts o f f i n anc i a l resources must be d i rected towards r a i s i n g the near-subsistence l eve l s o f consumption of a rap id ly growing populat ion. C lea r l y there i s a need f o r an integrated po l i cy of resource management and economic development whereby sustained growth w i l l be a t ta ined before the country ' s resources are exhausted. 16 The optimal rate of resource ext rac t ion and optimal a l l o c a t i o n of mineral revenues between consumption and investment should be simultaneously determined; the theory.of conservation i s subsumed in the theory of i n ve s t -ment, both invo lv ing the husbanding of resources f o r future use. For any act o f ex t rac t ion (and consumption) o f a depletable resource the soc iety en-joys an immediate benef i t f o r which an opportunity cost w i l l be paid in the fu ture . For any act o f investment in reproducib le c a p i t a l , the soc ie ty pays - 4 -an immediate opportunity cost in terms of lower current consumption in ex-change for the benefit of higher future consumption. Thus, optimality re-quires that the util ity of a unit of consumption sacrificed at one point dm time be balanced against the util ity of additional consumption gained at an-other point in time. Since mineral rights in developing economies are usually owned by the government which is also responsible for the country's economic development, attention is focused on the policies of the government in its dual role: (i) as the primary resource owner and hence entitled to resource rents and (ii) as the country's principal development agent and hence responsible for the allocation of these rents between consumption and capital investment. Gov-ernments in LDCs have either failed to recognize the need for an integrated resource and development policy, or,when they did, such policy was pursued in the context of five-year development plans. These plans, based on short-run price expectations and highly restrictive production technologies, have met with only limited success.1^ Because of the intertemporal nature of the problem, and the short-run fluctuations of mineral prices, an integratedfresource and development pol-icy should be formulated in the context of a long-run development plan based on: (i) society's intertemporal consumption preferences; (ii) long-run relative price expectations.1^9 ( i i i ) knowledge of the mining and non-mining production technologies; and (iv) information on the size and quality of the mineral (and other) resources. - 5 -Although i t i s the i n te rac t i on among these fac tors that determines the o p t i -mum p o l i c y , the fo l lowing points serve to i l l u s t r a t e the ro le of each f a c t o r . Long-run p r i ce expectations help in est imating the amount of mineral revenues expected to be received over the planning hor izon. The s i ze and the qua l i t y of the resource and the mining production technology serve to determine mining  cos t . F i n a l l y the intertemporal consumption preferences and the two prod-uct ion technologies are instrumental in determining the a l l o c a t i o n of net revenues between consumption and the two types of investment ( i . e . dm mining and non-mining reproducib le c a p i t a l ) . 1.2 Previous Studies 18 Previous studies on the potent ia l r o l e of mineral revenues in econ-omic development, with three important except ions, f a i l e d to recognize the s imultanei ty and intertemporal nature of the problem. They employed a s t a t i c ana l y t i c a l framework in analyzing what is b a s i c a l l y a normative dynamic pro-blem. Of p a r t i c u l a r importance i s the f a c t that they ignore the user cost a r i s i n g from the f i n i t enes s and non-renewabi l i ty o f ore depos i t s . User cost i s the intertemporal opportunity cost which a country should charge i t s e l f when determining the current rate of ext rac t ion in order to take into account the f a c t that the mineral extracted and exported today w i l l not be ava i l ab le in the future (see Scott [1967], He l l iwe l l [1974]). The three exceptions are Kendrick and Tay lor [1971], Samii Vajed [1975], Nagatani and Neher [1976]. A l l three studies have recognized the intertemp-oral nature of the problem. Nagatani and Neher's study involves the o i l pro-ducing developing countr ies but in the context of a world general equ i l i b r ium model; the LDCs ro le i s that of a monopolist (OPEC) invest ing the revenues from o i l exports in the world cap i t a l market rather than in t h e i r nat ional - 6 -economies. Kendrick and Tay l o r ' s i n t e r e s t centers on the a l l o c a t i o n over time of o i l revenues between consumption and investment in the domestic economy. They assume, however, that o i l export revenues are exogenous to the model, so that a resource sector as such does not enter in to the model and hence the problem of user cost does not a r i s e . Samii Vajed [1975], in a n e o c l a s s i c a l , planning model f o r Iranian o i l , 19 d id take in to account the user cost but ignored ext rac t ion cos t s . How-ever, not a l l gross revenues from mineral exp lo i t a t i on s - a re ava i l ab le f o r consumption and non-mining investment; only the surplus above the to ta l costs of e x t r a c t i o n , inc lud ing a l l labor costs and f u l l return on c a p i t a l , con-st i tutes, economic rent . In f a c t , most of the problems associated with r e -source - based indus t r ie s are ignored in Vajed's model. For ins tance, no cons iderat ion i s given to the problems of heterogeneous• qaailliiitjtiandtdff.miiin^--1-'' i sh ing a c c e s s i b i l i t y of the resource. As the mining production technology is completely ignored, problems o f f i xed versus var i ab le inputs and the sub-s t i t u t a b i l i t y between domestic and imported inputs cannot be handled. Fur-thermore, Vajed does not t e l l us how we might obtain an estimate of a country ' s mineral resource; he employs reserves estimates which are c l e a r l y inappropr iate in the case of heterogeneous mineral resources. F i n a l l y Vajed employs a h igh ly r e s t r i c t i v e funct iona l form (Cobb-Douglas) to represent the non-mining production technology. There i s , the re fo re , a need fo r ( i ) a theore t i ca l model of economic development through optimal mineral e x p l o i t a t i o n which incoporates mineral resource heterogeneity, var iab le ext ract ion costs and f i xed mining i nve s t -ment; and ( i i ) a numerical so lu t ion to such model based on n o n - r e s t r i c t i v e funct iona l s p e c i f i c a t i o n of the production technologies and proper e s t i -mation of the s i z e and qua l i t y of the mineral resource. - 7 -1.3 Scope and Out l ine of the Present Study Obvious ly, the above pro ject i s s u f f i c i e n t l y involved to warrant more than one study. Hence, the scope of the present thes i s i s l im i ted to what we consider to be the most unsat i s factory features of the received l i t e r a -ture: ( i ) the construct ion of the theore t i ca l model and der iva t ion of some q u a l i t a t i v e r e s u l t s ; ( i i ) the funct iona l s p e c i f i c a t i o n and est imation of mining and non-mining production technolog ies ; and ( i i i ) the measurement of the i n i t i a l stocks o f reproducib le and natural c a p i t a l , with p a r t i c u l a r em-phasis on the s i z e and q u a l i t y o f the mineral resource. Although the f u l l numerical so lu t ion of the model i s beyond the scope of the present study, the s p e c i f i c a t i o n s , and est imation of the technolog ica l parameters and i n i t -i a l condit ions undertaken in th i s study is a necessary step which must be completed before a numerical so lu t i on can be attempted. In developing the theore t i ca l model (Chapter 3) we draw on the Zambian 20 experience in copper mining and economic growth (Chapter 2) , and we make extensive use of a number of important advances in dua l i t y theory and i n t e r -temporal opt imiza t ion . The theore t i ca l framdwork is that of an optimal growth model which integrates the consumption-oriented cap i ta l model (Ramsey [1928]) and the exhaust ib le resource model (Hote l l ing [1931], Scott [1967] and o ther s ) , extended to the case of a heterogeneous mineral resource. The spec ia l features of the proposed intertemporal model are: ( i ) the d i s t i n c t i o n between mining and non-mining technology, between f i xed and va r i ab le inputs on the one hand, and between domestic and imported inputs , on the other. This second feature a r i ses from r e l a t i n g the model to developing countr ies where day-to-day mining operations are l e f t to fore ign f i rms , employing con-s iderab le amounts o f imported inputs . The re su l t i n g two-level (government 21 and f irm) opt imizat ion problem i s resolved by employing dua l i t y theory - 8 -(Diewert [1974]) to represent the mining technology by a va r i ab le cost func-.. tiiion. Using Shephard's [1953] lemma we der ive short-run cost-minimiz ing de-mand funct ions which are instrumental in formulating a two-level opt imiza-t ion problem into a s ing le model. Once the model i s constructed we employ the Maximum P r i n c i p l e of Potryagin e t . a l . [1962] to derive the optimal rCiiltes of resource ext rac t ion and revenue a l l o c a t i o n and the zero p r o f i t cond i t ions . The optimal t ime-path of per cap i ta consumption i s a l so der ived. These q u a l i t a t i v e resu l t s v e r i f y and extend e a r l i e r re su l t s of the received l i t e r a t u r e . Consider, as an i l l u s t r a t i o n , the zero p r o f i t cond i t ion fo r the mineral resource. The received theory f o r a homogeneous resource t e l l s us that the user cost (or dynamic shadow pr i ce ) o f the resource grows at the rate of i n t e r e s t . Having extended the conventional model to take account of population growth and se-quential dec l ine in resource q u a l i t y , we obtain the r e s u l t that the user cost o f the resource grows at the rate of i n te re s t plus the rate of popul -at ion growth minus the rate at which production cost r i se s due to the deple-t ion of better ores. Moreover, the use of dua l i t y theory enables us to r e -l a te the dynamic shadow pr ices of the various stocks to t h e i r s t a t i c shadow pr ices through the use of shares and e l a s t i c i t i e s and examine the i m p l i c -at ions of a p r i o r i r e s t r i c t i o n s on the production technolog ies. Having analyzed the ro le of production technologies we turn to t h e i r funct iona l s p e c i f i c a t i o n (Chapter 4) . This i s an important step towards a numerical s o l u t i o n , that has not been deal t with s a t i s f a c t o r i l y in the p e r t i -fiierjifet l i t e r a t u r e . F i r s t , we re lax the conventional assumption of unitary or constant e l a s t i c i t y of input subs t i tu t i on in the non-mining sector by s p e c i -fy ing a t rans log production func t i on . Second, we f i l l a gap in the l i t e r a -ture on cost fuct ions by developing the trans log var iab le cost funct ion f o U - 9 -Rowing Diewert 's [1974] genera l i za t ion of the va r i ab le p r o f i t f u n c t i o n . A l -though the constructed cost funct ion i s appl ied to the case of mining, i t i s general enough to be used f o r any technology invo lv ing f i xed and va r i ab le inputs . Most of the remainder of the thes i s (Chapters 5 through 7) i s devoted to the est imation of the t rans log va r i ab le cost funct ion and the t rans log production funct ion us ing, r e spec t i ve l y , Zambian data on copper mining and the rest of the economy. Zambia in comparison to other developing countr ies has r e l a t i v e l y extensive and r e l i a b l e data (though in raw form) which can be used f o r empir ica l work. A substant ia l port ion of the empir ica l cont r ibut ion of the present study i s the construct ion of appropriate data ser ies on inputs , outputs and rental pr ices dJor the mining and non-mining sectors of Zambia (Chapter 5). The primary purpose of these data i s t h e i r use in est imating the two a fo re -mentioned funct ions . However, p a r t i c u l a r care was taken to ensure con s i s t -ency of the data with the theore t i ca l framework of the intertemporal model so that they might be used in obtaining a numerical s o l u t i o n . In cons t ruct -ing these ser ie s we employ standard methods such as the perpetual inventory approach to cap i t a l s tocks , the D i v i s i a Index method of aggregation and the neoc las s i ca l theory of rental p r i ce s . The recent studies by Diewert [1975] [1977] and Woodland [1972] [1975] [1977] were extens ive ly consul ted. More-over, the Lasky [1950] approach of est imating the resource stock from the h i s tory of past production and reserves was extended and employed to obtain estimates of Zambian copper resource and grades. Chapter 6 i s devoted to econometric cons iderat ions such as the stochas-t i c : s p e c i f i c a t i o n of the est imating equations, the est imation techniques and hypothesis t e s t i n g . The empir ica l re su l t s are reported and in terpreted - 10 - . in Chapter 7. Various e l a s t i c i t i e s and s t a t i c shadow pr ices are computed, and a number of hypotheses r e l a t i n g to input subs t i tu t i on and technica l changes are tes ted. In Chapter 8 we return to the intertemporal model to examine how the estimated technolog ica l parameters might be used in ob ta in -ing a numerical s o l u t i on . The model i s re -wr i t ten in d i s c re te form and the so lut ion algorithm i s expla ined but a complete numerical s o l u t i o n , would require a s u f f i c i e n t l y involved s imulat ion pro ject to warrant a separate . study. F i n a l l y Chapter 9 assesses the overa l l resu l t s and discusses the scope f o r fu r ther research. There are three appendices: Appendix A describes the techn ica l as-pects of mining and ore treatment; Appendix B ind icates how some of the assumptions of the theore t i ca l model might be re laxed; and Appendix C con-ta ins some supplementary data tab le s . Inev i tab ly the present study has i t s l i m i t a t i o n s . In order to reduce the planning model to a manageable proport ion a number of r e s t r i c t i v e assump-t ions were made. F i r s t l y we adopted a rather narrow d e f i n i t i o n of economic development which i d e n t i f i e s development with sustained growth and ignores some s t ruc tu ra l and income-d i s t r ibut ion problems. Secondly, the subsistence sec tor , a common feature of most developriing countr ies i s not e x p l i c i t l y considered because of problems of measurement. This deprives development of part of i t s conventional meaning as a process through which the subsistence sector i s modernized. T h i r d l y , the issue of l inkages i s not considered here. While the l inkages between mining and other sectors are known to be weak, there i s the p o s s i b i l i t y of strong l inkages between the non-mining and the subsistence sec tor s . These may, in tu rn , a f f e c t the optimal a l l o c a t i o n of investment between the mining and the non-mining sec tor s . Fourth ly , the model cannot handle s i tua t i ons in which population growth i s not exogenously given but endogenously generated as a funct ion of the consumption l e v e l . F i n a l l y , i t must be noted that we have based our planning model on the rather simple case of Zambia. Other mineral-dependent economics such as Ch i le and L i b e r i a present add i t iona l compl icat ions. - 12 -FOOTNOTES TO CHAPTER 1 1. See fo r instance: N icho l l s [1963]; Fei and Ranis [1964]; Chenery and McEwan [1966]; and Drake [1972]. 2. See Mikesel l [1970:3]. 3. As ind ica ted l a t e r in th i s chapter the var i ab le cost funct ion was deve l -oped from the example of Diewert 's [1974] genera l i za t ion of the va r i ab le p r o f i t f unc t i on . It i s in f ac t a spec ia l case of the l a t t e r . The ad-v ice of Professor Diewert in th i s respect i s g r a t e f u l l y acknowledged. 4. An economy is sa id to be in a s ta t ionary s tate when i t experiences a (near) zero growth rate of per cap i ta income. 5. This i s Adelman's [1971:1] d e f i n i t i o n of economic development. It i s widely accepted among development economists but i t must be noted that there are numerous d e f i n i t i o n s of development ranging from those which i d e n t i f y development with growth to those which combine so many v a r i -ables that become de sc r i p t i ve rather than a n a l y t i c a l . Throughout th i s study we adopt Adelman's d e f i n i t i o n , which i d e n t i f i e s development with per s i s tent endogenously generated growth. 6. Adelman [1971:1]. 7. For instance development of subst i tutes and a change in technology rend-ered Chi lean n i t r a te s unpro f i t ab le . 8. It i s assumed that the pr ices o f a l l other commodities ( inc lud ing im-ported goods and mining inputs) remain constant. 9. This i s p a r t i c u l a r l y true of copper (see Table I M below). 10. OPEC succeeded in r a i s i n g the p r i ce of on'il by 700 percent in a 6-year per iod (1970-76) and mu l t ip l y ing the o i l revenues of i t s members by a f ac to r of 10. This i s based on f i gures reported by E l l i s [1974:4-46]. 11. See footnote 53 to Chapter 2. 12. See Brown and Bu t t l e r [1968]. 13. Dependence in terms of savings and fore ign exchange. 14. Investment in reproducib le cap i ta l i s a lso warranted on grounds of i n t e r -generational equi ty . See Solow [1974]. 15. Diminishing a c c e s s i b i l i t y and f a l l i n g grades are d i r e c t e f fec t s o f cumu-i-aMvee product ion. The i m p l i c i t assumption (made e x p l i c i t l a t e r ) i s that best grades are mined f i r s t . 16. Here opt ima l i ty i s used in a somewhat loose sense. Later on, in Chapter 3, op t ima l i t y i s p rec i se l y def ined in terms of maximizing a ce r ta in - 13 -soc i a l welfare f unc t i on , over a given planning horizon subject to approp-r i a t e cons t ra in t s . 17. See f o r instance the experience of Zambia as descr ibed in Chapter 2, be-low. Note that by h ighly r e s t r i c t i v e technologies we mean the use of f i xed cap i ta l /ou tput and c a p i t a l / l a b o r r a t i o s . 17ja. By r e l a t i v e p r i ce expectations we mean p lanner ' s expectations of mineral pr i ces r e l a t i v e to a l l other goods produced or imported by the country concerned. These r e l a t i v e pr ices determine what the mineral resource i s worth to the country. 18. For ins tance, see c o l l e c t i o n s of studies in Mikesel l [1970] and Seidman [1975]. 18a. Professor Bradley has brought to my at tent ion some recent studies on OPEC by.Cremer and Wettzman [1976] and Ezzat i [1976]. While these s tud-ies do take account of the user cost of o i l thedir ob jec t i ve i s to p red i c t the monopoly p r i ce of o i l rather than analyze the development potent ia l of the o i l exporting count r ie s . 19. Vajed [1975] was, in a sense, j u s t i f i e d in ignoring ex t rac t ion costs in the case of o i l : in 1975 the ext rac t ion cost of Iranian o i l was about $0,053 compared to i t s market p r i ce of $9.60. This i s not the case with copper and other minerals : the production cost o f Chi lean copper in 1970 was about 30.88 cents per pound compared to a p r i ce o f 50 cents per pound (See Seidman 1975, p. 5). 20. In developing the model Zambia i s used as a working example of a mineral dependent economy because of the s i m p l i c i t y af forded by i t s extreme de-pendence on a s ing le minera l . Not withstanding the complexity of other mineral exporters , such as C h i l e , i t can be sa id that the model has gen-era l a p p l i c a b i l i t y to any mineral dependent economy. However, Zambia best approximates the assumptions of the model. 21. See sect ion 3.5 fo r d e t a i l s . CHAPTER 2 COPPER MINING AND ZAMBIAN ECONOMIC GROWTH: THE NEED FOR LONG-TERM RESOURCE AND DEVELOPMENT PLANNING Among a l l (non-petroleum) minera l -export ing countr ies Zambia presents the most extreme example.of dependence on a non-renewable resource not only f o r i t s economic development but f o r i t s bare surv iva l as w e l l . The copper industry contr ibutes almost one ha l f of the Gross Domestic Product, more than 90 percent of Zambia's to ta l exports and fore ign exchange earnings, and about 16 percent of non-subsistence employment. 1 For th i s reason, Zambia was chosen as a case study. It i s hoped that th i s extreme dependence w i l l sharpen the re su l t i n g imp l i ca t i ons . Moreover, in comparison to either minera l -export ing developing countr ies Zambia has r e l a t i v e l y extensive and r e l i a b l e data f o r empir ica l work. The purpose of the present chapter is t h ree fo l d : f i r s t l y , to present a b r i e f h i s t o r i c a l account of the Zambian copper industry and the res t of the economy; secondly, to review the Zambian p o l i c i e s of economic develop-ment through mineral e x p l o i t a t i o n ; and t h i r d l y , drawing on the Zambian ex-per ience, to demonstrate the need fo r minera l -export ing developing countr ies to formulate resource and development p o l i c i e s in the context of a long-term development p lan. 2.1 General Background Zambia i s a landlocked country in Southern A f r i c a covering 290,386 square mi le s , and sharing f r o n t i e r s with Rhodesia, Mozambique, Angola, 2 Botswana, Namibia, Za i re , Tanzania and.Malawi. Admin i s t r a t i ve l y , the coun-t r y i s d iv ided into e ight provinces: Copperbelt, Cen t ra l , Eastern, North-western, Luapula, Northern, Southern and Western. The cap i t a l c i t y i s Lusaka and other major c i t i e s are Kitwe, Ndola, Mufu l i ra and Luanshya. The popula-t i on was estimated at 4.7 m i l l i o n in 1974 with an annual rate of growth of 3 about 2.5 percent. H i s t o r i c a l l y the country was f i r s t organized under the ru le of the B r i t i s h South A f r i c a Company (BSA Co.) es tab l i shed by Cec i l Rhodes in the l a te 19th century. In 1911 the country was un i f i ed as one en t i t y under the name Northern Rhodesia, and in 1924 the admin i s t rat ive control was handed over to the B r i t i s h Colonia l O f f i c e . In 1953 a federa -t ion was es tab l i shed between Northern Rhodesia (now Zambia), Southern Rhodesia (now Rhodesia) and Nyasaland (now Malawi), which l a s ted f o r a decade. In October 1964 Northern Rhodesia gained i t s f u l l independence 4 under the name Republic of Zambia. 2.2 The Zambian Copper Industry, 1920-1964 Systematic mineral exp lorat ion in Zambia did not commence un t i l the mid 1920's and large sca le copper mining began only in the ear ly 1930's. The main reason f o r the delayed development of the copper industry in Zambia was the unattract iveness of i t s oxide ores of 3 to 5 percent average grade as compared to neighboring Katanga where the average grade of the ore was as high as 15 percent. This coupled with the lack of funds on the part of the BSA Co. hindered the large sca le exp lo ra t i on , needed to uncover the import tant geolog ica l information which was to make Zambia the la rges t copper exporter in the world. At moderate depths, the oxide ores were changing to sulphide ores of the same average grade; sulphide ores unl ike oxide ores , do 5 not , requ i re the co s t l y leaching process. Baldwin [1966] a t t r i bu te s the systematic explorat ion of the 1920's to the fo l lowing reasons: (a) the r i s e in copper pr ices under the pressure of the new demand generated by the automotive and e l e c t r i c a l indus t r ies (b) the - 16 -improvement of the technology of concentrating ore (especially the perfect-ion of the "flotation" method) and, most importantly, (c) BSA Co.'s new pol-icy of granting exclusive prospecting rights over large areas to financial-l y powerful companies. Table I below indicates the exclusive prospecting licences granted init ia l ly to relatively small concession companies but grad-ually passed on to two giant mining corporations: the Anglo-American Corpor-ation of South Africa (AAC) and the Rhodesian Selection Trust (RST) in which the American Metal Climax of New York (AMAX) had controlling interests. Following these concessions an unprecedented exploration campaign was launched between 1924 and 1940, which led to the discovery of the largest Zambian orebodies: Rhokana in 1924, Nchanga and Roan Antelope in 1926, Min-dola in 1927, Mufulira and Nchanga West in 1928. Discovery of smaller ore-bodies, such as Chibuliiama or new extensions of the existing ones continued during the decade 1930-39. Drysdall [1972] reports that from the total con-cessionsarea of 200,000 square miles, "156,000 square miles - 54 percent of the country - were prospected in detail, that is by systematic straight-line foot traverses usually spaced at quarter mile intervals. All outcrop data were recorded, and mineral occurences were followed up by pitting, trench-ing and diamond dr i l l ing . . . " 7 In the late 1920's development work was taking place on what came to be Zambia's four major mines: Rhakana and Nchanga controlled by AAC and Roan Antelope and Mufulira controlled by RST. Production on a large scale, however, did not commence until the early 1930's. Before that date no sig-nificant copper production had taken place except for the small-scale mining operation at Sable Antelope and Silver King Mines and the unprofitable pro-duction of 22,000 metric tons at Bwana Mkubwa open pit. g The f irst large scale production occurred in 1932 when Roan and Rhokana commenced mining operations and the two alone produced in one year a to ta l o f 60,000 metric tons of copper, twice the cumulative copper product tion of Northern Rhodesia fo r the en t i r e per iod 1900-1931. This ra i sed N. g Rhodesia's share in the world market from less than one percent in 1931 (with 9,000 metr ic tons) to about 7 percent in 1982. The fo l lowing year Mufu l i ra a l so commenced production and Northern Rhodesia's share of the world market was ra i sed to 10 percent and a year l a t e r to 14 p e r c e n t . 1 0 Nchanga mine, the r i che s t in copper content (4.66 percent) among the major mines, d id not enter production unt i l 1939 but, then, i t experienced such a f a s t rate of growth that by 1969 nit was producing one quarter of N. Rhodesia's to ta l copper product ion. Two more mines of l e s se r s i gn i f i c ance Chibuluma of RST and Bancroft o f AAC opened in 1956 and 1957 re spec t i ve l y (see Table II below). No other s i g n i f i c a n t addit ions were made with in the per iod 1957-1964 except fo r extensions of the ex i s t i ng mines and expansions of the m i l l i n g capac i ty . A r e f i n i n g f a c i l i t y , the Rhokana Copper Ref iner ies L td . , opened in 1935 and was j o i n t l y and evenly owned by Nchanga and Rhokana of the AAC. The RST Group es tab l i shed i t s own re f i ne ry f a c i l i t y , Ndola Re-f i n e r i e s L td . Table II below, gives discovery dates o f main orebodies, and copper production and ore reserves at the s t a r t of mining operations in the seven largest mines. Yearly f i gures on ore reserves, average grade and ore h o i s t -ed f o r i nd i v idua l mines are given in Table XLVI of Appendix C. The corres - . pcnMing aggregate f igures f o r Zambia are found in Tables XIV and XV of Chap-ter 5. Table XV a l so contains data on aggregate copper product ion. The rap id expansion of the copper industry described so f a r required enormous f i n a n c i a l resources fo r cap i t a l investment in s t ructures and equip-ment such as mining sha f t s , m i l l s , concentrators and r e f i n e r i e s , mining towns - 18 -and roads, moving and handling equipment. Although at the s t a r t , the com-panies ra i sed the money abroad through t h e i r parent f i rms , they became i n -creas ing ly able to f inance t h e i r expansion from reta ined p r o f i t s . By 1944 the cumulative net investment (cap i ta l stock) in mining had reached the mag-nitude of K160,000,000 in 1967 p r i c e s 1 1 (see McPhesson [1976a:10]). Gross f i xed cap i t a l formation in mining continued into the 1950's at an increas ing ra te . From the 1945 leve l of K6,000,000 gross investment reached a peak in 1957 of K68,230,000 (both in 1967 p r i c e s ) . Subsequently, gross investment fol lowed a downward trend f a l l i n g to less than ha l f the 1957 f i gure (K32.531,000) by the time of Zambian independence in 1964 (see McPherson [1976a:13]). On the demand s i d e , the period 1930-1964 i s character ized by r i s i n g world copper consumption at an average rate of 4 percent annual ly. Pr ices 12 f luc tua ted widely but had an upward t rend. When development at the four la rges t Zambian mines was taking place in 1929, the average p r i ce of copper was Kl64 per metric ton. Three years l a t e r , 1932, when production began, the p r i ce dropped to K50 per metric ton, despite an agreement between major producers to cut production by 25 percent. This led to the co l lapse of the copper c a r t e l , Copper Exporters Inc. The p r i ce of copper did not return to i t s 1929 peak un t i l 1947 when i t jumped to K257 per metric ton from Kl51 in 1946. Then i t continued i t s upward trend un t i l 1955 when i t reached a new peak of K637 per metr ic ton. In 1958 i t f e l l again to a low of K311 from which i t s ta r ted r i s i n g unsteadi ly to a t t a i n i t s h i s t o r i c a l l y highest peak of almost K1000 per metric ton by 1970 (see Table X). Zambian copper exports fol lowed the rate of copper production c l o se l y s ince domestic copper consumption was neg l i g i b l e and no s i g n i f i c a n t s tock-- 19 -1 3 p i l i n g took p lace. The value of Zambian exports rose from K14,000 in 1924 to K492,000,000 in 1964; that i s , from less than 2 percent to more than 90 percent o f to ta l exports. By 1964 c lose to 10 m i l l i on s metric tons of r e -f i ned copper were extracted and exported. - 20 -TABLE I PROSPECTING LICENCES, ZAMBIA, 1922-30 Year 1922 1923 1925 1930 Or ig ina l Concession Companies Copper Ventures Limited II n II Luangwa Concessions (N.R.) Limited II M H Total Concession Area 50,000 square miles 15,000 " 85,000 " 50,000 " 200,000 square miles Source: Compiled foirm? data reported in Drysdal l [1972: 56-61] TABLE II ZAMBIAN COPPER MINES: DISCOVERY AND INITIAL CAPACITY Mine Date o f a Discovery S tar t o f Mining Operations Ore Reserves Average Date (metric tons) grade (percent) Annual CAnnualOutpu. CoppercOutput (metric tons) Roan (RST) 1926 1932 88,935,000 a 3.44 a 31,242 Rokana (AAC) 1924 1932 104 363,000 a 4.00 a 29,017 Mufu l i ra (RST) 1928 1934 95,288,000 a 4.41 a 5,653 Nchanga (AAC) 1926 (1928) 1940 118,338,000 a 4.66 a 13,976 Chibuluna (RST) 1939 1956 6,624,000 5.23 4,224 Bancroft (AAC) 1952-1953 1957 93,103,000 3.89 43,502 Chambishi (RST) 1947 (1956) 1965 31 ,760,000 3.37 2,977 a. Source: Drysdal l [1972: 58-65, 69] b. Although operations may have s tar ted the preceding year we record the f i r s t year at which f u l l production was a t ta ined . c. Dates r e f e r to year ending June 30 except f o r Nchanga and Bancroft fo r which the year ends March 31. For ins tance, 1932 is most appropr ia te -ly read as 1931-32. General Source: Table XLVI of Appendix C. - 21 -2.3 Economic Conditions in The Non-Mining Sector, 1920-1964 Before the r i s e of the copper industry in the ear ly 1930 1s there was only a neg l i g i b l e commercialized sector (mainly lead , z inc and a g r i c u l t u r a l exports) organized and run by European s e t t l e r s . The res t of the economy cons is ted of subsistence a g r i cu l tu re employing p r im i t i ve labor-and- land i n -tens ive techniques and no machinery. The economic condi t ions o f the A f r i c an population during the 1920's were best descr ibed by Baldwin [1966]: Economic l i f e f o r most of the nearly 1 m i l l i o n A f r i cans was not too d i f f e r e n t from the p r i m i -t i v e s tage. . . the economy was s t i l l such that more than 96 percent of the population l i v e d in the rura l subsistence s e c t o r ! 5 . . . . Ca t t le were the only important l i q u i d asset that an A f r i c an could accumulate in the subsistence economy and were regarded as the main measure o f wealth.16 The massive economic a c t i v i t y that fo l lowed the discovery of the Cop--pafibelt in the l a te 1920's soon transformed Northern Rhodesia in to one of the most r ap id l y growing economies in the world as r e f l e c t e d in the average an-nual rate of 8.5 percent at which the gross domestic product has grown s ince 1 9 4 5 . ^ This meant that by the time of independence Northern Rhodesia was enjoying a per cap i ta GDP of K132 a year , the highest in A f r i c a i f we ex-clude South A f r i c a . But to what extent did th i s spectacular growth mean any advancement of the country ' s development process over the subsistence con-d i t i ons of 1920's? Very l i t t l e i s known about the non-mining sector o f Northern Rhodesia during the per iod 1930-44 s ince National Accounts were not compiled before 1938 and there ex i s t npore l i ab le f igures f o r the war years . Hence, we w i l l concentrate on the post-war per iod 1945-64. Stimulated by post-war recons t ruc t i on , the expansion of the e l e c t r i c a l - 22 -i ndus t r i e s and the co l d war of the 1950's, copper demand was s u f f i c i e n t l y strong to keep copper pr ices r i s i n g f o r an en t i r e decade: from Kl22 per ton in 1945 the p r i ce of copper reached an average maximum of K637 in 1955. Thus, with the end of the war, "the Copperbelt entered in an era of unpara l le led 18 prosper i ty " during which i t could be reasonably expected to play the ro le of a leading sec to r .o f economic growth s t imulat ing the development of the non-mining sector o f the Northern Rhodesian economy. Indeed the Gross Domestic Product (GDP) at current pr ices rose at an 19 average of 12.3 percent over the per iod 1945-54. This unusually high rate of growth, however, was sustained by purely exogenous fo rces : s p e c i f i c a l l y , the strong world demand fo r copper which led to r i s i n g copper pr ices over the en t i re per iod (see Table X). The f a i l u r e of the economy to a t t a i n " s e l f -sustained growth" became apparent in the per iod 1955-64: f a l l i n g or f l u c t u -at ing pr ices led to negative or near zero rates of growth, o f real GDP o r i g -ina t ing in mining and with some lag to negative or near zero growth rates f o r the aggregate real GDP (see Table I I I). The only exceptions were the years 1959 and 1964 when remarkable growth spurts were achieved as a r e s u l t o f a sudden r i s e in copper pr ices in 1959 and a new upward trend i n i t i a t e d in 1964. The average annual rate of growth f o r the en t i r e per iod (1954-64) was 5.8 percent due mainly to these two years . This sporadic growth can hardly be in terpreted as achievement of "economic development" in Adelman's [1971] d e f i n i t i o n of the term (see Chapter I). The heavy re l i ance of Northern Rhodesia on copper mining i s revealed 20 by more than one s t a t i s t i c . The mining sector dominated every aspect of the economy, accounting f o r as much as 50 percent of GDP i n current p r i c e s , more than 90 percent o f exports , and about 17 percent of employment (see - 23 -TABLE III COPPER PRICES, GROWTH RATES AND MINING SHARE IN GDP, EXPORTS AND EMPLOYMENT, ZAMBIA, 1954-64 YEAR PCU GYM GYZ GYN SYM SXM SNM 1954 454 - - - 53.9 - 18.0 1955 637 2.1 7.1 17.5 56.3 95.0 17.0 1956 609 9.8 11.4 18.8 53.6 94.0 17.5 1957 388 -24.3 -9.3 10.9 40.0* 92.0 17.0 1958 311 -10.2 -2.9 2.9 33.6 92.0 15.0 1959 412 79.2 37.2 5.0 45.2 93.0 16.5 1960 430 -3.9 -1.5 1.7 47.8 94.0 17.0 1961 404 -6.7 -1.6 2.6 44.7 93.0 17.6 1962 410 -2.5 0.5 1.1 43.8 92.0 17.7 1963 410 0.8 1.1 1.7 42.6 92.0 18.0 1964 434 14.1 11.4 9.1 45.8 92.0 17.6 A l l f i gures except PCU, are expressed in percentages. PCU : copper pr ices (Kwachas per metr ic ton) GYM : growth rate of real (1967 pr i ces ) Gross Domestic Product (GDP) at f ac to r cost ( f . c . ) o r i g i n a t i n g in the mining sec tor . GYZ : growth rate of aggregate real GDP at f . c . GYN : growth rate of real GDP at f . c . o r i g i na t i n g in the non-mining sector . SYM : share of mining in aggregate GDP at f . c . in current p r i c e s . SXM : share of mining in value of aggregate exports. SNM : share of mining in non-subsistence employment. Sources: PCU: Table X, GYM, GYZ, GYN and SYN: ca l cu l a ted from data given in Table XXIX, XLI, arid XLII, SNM: ca l cu l a ted from data given in Tables XI and XLII I, and SXM: Mining Yearbook of Zambia (various) i s sue s ) . - 24 -Table II I). Over the 11 years between 1954 and 1964 there appeared no signs i nd i c a t i n g a reduct ion o f t h i s extreme dependence on a s ing le v i r t u a l l y un-con t ro l l ed industry ; and although populat ion and potent ia l labor force were rap id l y growing, the copper industry f a i l e d to generate add i t iona l employ-ment e i t h e r d i r e c t l y in mining or in other i ndu s t r i e s . The only indus t r ie s of any s i gn i f i c ance besides mining were those highly dependent on the mining indust ry . Although most of the inputs into mining ( inc lud ing part o f the labor force) were imported, l o c a t i o n - s p e c i f i c indus t r ie s such as e l e c t r i c power, r a i l t r anspor ta t ion , bu i l d ing and con-s t ruc t i on and government serv ices were st imulated by the expansion of the copper i n d u s t r y . 2 0 3 These i n d u s t r i e s , however, were t i g h t l y l inked to the mining industry and located in the mining areas (mainly the Copperbelt prov-ince) and along the " l i n e of r a i l " that runs from the Southern Rhodesian border through the cap i t a l Lusaka to the Copperbelt. This was a v i v i d ind i t Nation of the narrowness o f the growth base achieved. The gravel and earth roads that connected the r a i l with the remote A f r i can v i l l a g e s were equal ly c l ea r i nd i ca t ions of the weakness of l inkages between mining and the subs is -encee sec tor s . In the words of Baldwin [1966]: A modern, h ighly mechanized industry has been . superimposed upon a r u r a l , backward economy, 72 percent of whose populat ion s t i l l i s engaged in subsistence p roduc t i on . . . apparently without s t imula t ing development in the l a t t e r s e c t o r . 2 1 Besides the f a i l u r e of the copper industry to st imulate widespread development, there i s one fu r ther reason why the growth f i gures even in per cap i ta terms shouilld not be in terpreted as achievement of economic develop-ment. Although income per cap i ta at the time of independence [1964] was among the highest in A f r i ca -about K70 or Kl40 a year - th i s says very l i t t l e - 25 -about the l i v i n g condit ions of at l ea s t three quarters of the populat ion "because the f i gure includes the earnings o f non-Afr icans which were among 22 the highest in the world" Martin [1972] estimates that f o r probably 70 per-cent of the people the per cap i ta annual cash income was no more than 10 to 20 Kwachas. 2 3 The mining r o y a l t i e s were paid to the B r i t i s h South A f r i c a Company, (BSA Co.) which had exc lus ive ownership of the m inera l . r i gh t s . It was not unt i l 1950 that the BSA Co. agreed to pay one f i f t h of the r o y a l t i e s to the Northern Rhodesian government. Even with only one f i f t h of the r oya l t i e s by 1953 the government was able to ra i se the Development Budget fo r the decade 1947-57 by K41 m i l l i o n ; but, the European bias of the development plan and the subsequent federat ion v i r t u a l l y n u l l i f i e d any benef i t s to Northern Rhodesia. The Federal government s i tua ted in Sa l i sbury reta ined a strong European b ias . With most Europeans l i v i n g in Southern Rhodesia, and with the bulk of government revenues, generated in the Northern Rhodesia the Fed-erat ion resu l ted in a net drain of copper revenues from the North to the South. Martin [1972] re fers to an estimate of a to ta l tax loss to Northern Rhodesia o f 97 m i l l i o n and an increase in nat ional debt by 75 m i l l i o n during 24 the federal per iod . The independence of Zambia in 1964 marked a new era in the country ' s development prospects. 2.4 Development Planning and Mineral Resource Management in Zambia, 1964-74 Upon independence, Zambia i nher i ted from the co lon ia l past aTl the features of an underdeveloped country. The Report of the U.N./E.C.A./F.A.O. Economic Survey Mission on the Economic Development of Zambia, 1964, summar-ized the l i v i n g condit ions in Zambia as fo l lows: "The great major i ty o f the people in Zambia are poor, under-educated ( i f not i l l i t e r a t e ) , and un-- 26 -25 hea l thy. " Although th i s could have been sa id f o r more than ha l f the coun-t r i e s in the world, i t appears a p r i o r i su rp r i s i ng f o r a country with less than 4 m i l l i o n people and more than 30 years of h i s tory in hosting one of the most l u c r a t i v e indus t r ies enormous in s i ze even by world standards. A l -though more than K600 m i l l i o n (1967 pr i ces ) was invested within Zambia and more than 10 m i l l i o n tons of copper exported, the loca l development which had been achieved was meagre: By independence manufacturing cons t i tu ted about 5.4 percent of Gross Domestic Product and commercial a g r i cu l tu re about 2 percent, both at f ac to r cos t . There were only about 1,000 secondary 26 school graduates and less than 100 Af r i cans with some un iver s i t y educat ion. But not only ' l i a b i l i t i e s ' were i n h e r i t e d . There were a l so some ' a s set s ' which caused Zambia to d i f f e r from other developing countr ies emer-ging as independent nat ions. Although the supply of phys ica l and human cap-i t a l was extremely l i m i t e d , the supply of f i n a n c i a l cap i t a l was p o t e n t i a l l y un l imi ted, despite a nat ional debt of 194 m i l l i o n Kwachas i nhe r i t ed from the 27 federal years . Zambia's s i x major copper mines were in f u l l operat ion with an annual production capac i ty of about 650,000 metr ic tons of re f ined copper, 28 and ore reserves of 658 m i l l i o n metr ic tons of 3.56 percent average grade. The copper companies "had long s ince paid fo r t h e i r o r i g i n a l investments out of t h e i r p r o f i t s , and continued to expand t h e i r output through reinvestment 29 and borrowed funds." Copper pr ices were on a new upward trend and more than doubled with in f i v e years o f independence. Furthermore, there were the s k i l l e d expa t r i a te s , and,some Af r i cans who had acquired s k i l l s by working in mines and on European farms. But most importantly there was substant ia l (though not adequate) i n f r a s t ruc tu re e s p e c i a l l y ra i lways , roads and power supply st imulated by mining. - 27 -The task confronting the f i r s t national government was to use the 'assets ' optimally to compensate for the ' l i a b i l i t i e s ' of more than 80 years of co lonia l rule of one sort or another. Copper mining was the only source of foreign exchange and savings, the major source of government revenues,and a substantial source of employment. And, as such, i t represented the sole foundation on which to base an intensive development strategy. The copper industry,, despite i t s past f a i l u r e to stimulate economic development, was, thus, ca l l ed upon again to serve as a leading sector; but, th i s time, within the context of an integrated resource and development plan. On the day of independence President Kaunda declared: Time and again we have made i t c lear that i t i s our intent ion to put th i s country on the road of progress through intensive eco-nomic a c t i v i t y based on our r i d i copper mines... 30 The rapidly growing population and gradually depleting mineral resour-ces ca l led for the design of a consistent pol icy of mineral resource manage-ment and economic development whereby sel f -susta ined growth of per capita i n -come would be attained before the country 's resources were exhausted. In the words of President Kaunda "gett ing the roof up before the foundations c o l l -31 apse." The mineral r ights and taxat ion, public expenditure and external trade control were the government pol icy instruments at the time. The found-ations of the government development pol icy were l a i d by the Seers Report op [1964] which outl ined the strategy of the Transit ional Development Plan (TDP) 1964-1966 and the F i r s t National Development Plan (FNDP) 1966-1970. The underlying philosophy of the Seers Report was d i v e r s i f i c a t i o n of the copper-based economy through import subst i tut ion po l i cy . Adhering to th i s po l icy the FNDP set the fol lowing objectives for the period 1966-70: - 28 -( i ) a target rate of growth f o r Gross Domestic Product of 11.7 percent per annum and an increase of per cap i ta income from K122 in 1964 to K200 by 1970; ( i i ) target rates of growth f o r manufacturing and a g r i c u l t u r a l output of 14 and 9 percent re spec t i ve l y to d i v e r s i f y the copper based economy; 33 ( i i i ) a target increase of wage employment by 100,000 jobs between 1966 and 1970; ( iv) a substant ia l decrease in both the r e l a t i v e and absolute share of non-Afr icans in wage employment e s p e c i a l l y in the copper industry by rep lac ing as many as poss ib le by Zambians; (v) a target annual growth rate f o r gross f i xed cap i t a l formation of 20 percent; cap i t a l investment of K858 m i l l i o n was planned, o f which K564 was to be provided by the government and K294 m i l l i o n by the pr iva te sec tor ; (v i ) target rates o f growth of pr iva te and government consumption of 12.4 percent and 18.2 percent r e s p e c t i v e l y ; ( v i i ) target rates fo r annual copper production ranging from 858,000 metr ic tons (high target) to 752,000 (low t a r g e t ) , and govern-ment revenues from mineral tax of the" order of K766 m i l l i o n (62 percent of the to ta l government revenues); The l a s t ob jec t i ve was an e x p l i c i t recogni t ion by the p lanners, that " cap i t a l investment ava i l ab le to the Government fo r f inanc ing the F i r s t Nat-ional Development Plan is determined almost e n t i r e l y by revenues obtained 34 from copper mining o p e r a t i o n s . . . " In the context of th i s ob jec t i ve the planners recognized the need fo r an optimal mineral resource management by suggesting (a) adoption of marketing and p r i c i n g p o l i c i e s to ensure an i n -creas ing market share f o r Zambian copper (b) development of ore bodies un-used at the time and of mineral resources in general " i n such a manner as to 35 guard against t h e i r rap id dep le t i on . " An eva luat ion of the overa l l performance of the FNDP (and TDP) i s pre-sented in the Table IV parts o f which are reproduced from the Second National Development P lan ' s review of the FNDP's performance: - 29 -or , a) The Gross Domestic Product (1964 pr i ces ) increased by 83 percent over the per iod 1965-1970 at an average annual rate of 10.6 percent or 8.2 percent per c a p i t a . Although th i s growth f a l l s short by 1.1 percent of the target growth rate of real GDP of 11.7 (see Table IV, rows A l , A2 and A3) i t compares favorably with the pre-independence per iod 1955-1964 where real GDP (1954 pr i ces ) grew only by.66 percent at an annual rate of 5.8 percent or at 2 percent in per cap i ta terms. As the Zambian planners admitted, in review-ing the performance of the FNDP, ha l f o f th i s growth (1965-1970) was the r e -s u l t o f unexpectedly high copper p r i c e s , which reached K1000 per metric ton by 1970. Had the copper pr ices remained at K690 per metr ic ton projected by the plan real GDP would have increased only by 6 percent per annum, only s l i g h t l y higher than the 5.2 percent of the federal years . b) Gross f i xed cap i t a l formation increased at an average annual rate of 21.0 percent, surpassing the planned rate by 1.0 percent (Table IV, rows B l , B2). The share of cap i t a l investment i n . GDP rose from 16.3 percent in 1964 to 27.8 percent in 1970 (row B4) which i s " i n d i c a t i v e of the f a c t that the country i s introducing s t ruc tura l changes in her investment patterns that are es sent ia l f o r promoting sustained soc ia l and economic develop;-m e n t . " 3 6 c) The average annual rate of growth of rea l pr iva te consumption f e l l short of the target rate o f 12.4 percent by 4.7 percent (see Table IV, rows Cl,and C2). This was pa r t l y a t t r i b u t e d to an increase in personal remit -tances abroad from K11.3 m i l l i o n in 1967 to K54.4 m i l l i o n in 1970. S i m i l a r -ly real government consumption grew by only 10.6 percent annual ly instead of the planned 18.2 percent. d) Non-subsistence employment f e l l short o f the target of 407,000 by 17,000 jobs (see Table IV row Dl) due mainly to a l a rger r i s e in real wages - 30 -than a n t i c i p a t e d . The mining sector exceeded i t s target employment whi le 37 the non-mining sector f e l l behind. The 'Zambianizat ion ' po l i c y was f a i r l y successful i n reducing the share of non-Afr ican employment from 13.3 percent in 1964 of the to ta l non-agr i cu l tura l employment to 7.7 in 1970. In the edu-cat iona l f i e l d the number of secondary school graduates increased by 600 percent over the 1964 leve l and the f i r s t Un iver s i ty was es tab l i shed with enrollment of 1 ,200. 3 8 e) The ob ject i ve of reducing Zambia's dependence on copper mining through d i v e r s i f i c a t i o n met with only l im i ted success. Of the twin elements of s t ruc tu ra l change, manufacturing and a g r i c u l t u r e , the former exceeded the target rate of annual growth by 3 percent, but, the l a t t e r f e l l behind i t s target growth rate by 5.7 percent, remaining v i r t u a l l y stagnant in per cap-39 i t a terms. Since manufacturing i s s t i l l a very small sector o f the econ-omy-its cont r ibut ion to GDP at f ac to r cost was hardly 8 percent in 1970 -i t s impressive growth d id not change the dependence of the economy on the mining sector not iceab ly . Table IV (rows E1-E4) ind icates that the mining sector cont inued, over the per iod 1964-1970, to account f o r more than 40 percent o f the GDP at f ac to r cos t , more than 50 percent of the Government revenues, more than 90 percent of to ta l exports , and about 16 of A f r i can wage employment. The past domination of the economy continued, pa r t l y due to very high copper pr ices and par t l y to a very modest success of the FNDP in d i v e r s i f y i n g the economy. Capi ta l formation was the only aspect in which s i g n i f i c a n t progress was made towards increas ing the share o f the ,non-mining sector . f ) Despite the extremely favorable copper pr ices the mining sector r e -mained v i r t u a l l y stagnant. In f a c t , copper output decreased from 695.7 metric tons in 1965 to 684.1 metric tons in 1970. While th i s was pa r t l y due - 31 -to an accident at the Mufu l i ra mine, the overa l l Copperbelt production grew only by 10 percent during 1965-69 (according to the most op t im i s t i c s t a t -i s t i c s 4 0 ) compared with 21 percent i n 1960-64 and 72 percent i n 1955-60. 4 1 This(meagre) growth of output was due par t l y to the opening of a new mine, Chambishi, in 1965, and a modest growth in the e x i s t i n g RST mines, whi le the production of the Anglo-American group f e l l by 2.4 percent over the per-42 iod 1965-69. While output f e l l short o f even the low target o f the p lan , as a r e s u l t of the unexpectedly high copper pr ices the to ta l contr ibut ion of income tax (or i g inated mainly i n the copper i ndus t ry ) , mineral r o y a l t i e s and copper export tax, exceeded the target of K768 m i l l i o n by 47.9 percent. As ea r l y as 1968 - two years before the F i r s t National Development Plan expired - the Zambian Government r e a l i z e d that despite the favorable copper pr ices the P lan ' s targets were not going to be met. Pres ident Kaunda sever-e ly c r i t i c i z e d the mining companies f o r the lack of mine development s ince 43 independence. The fore ign companies responded by blaming t h e i r poor per-formance on the complicated taxat ion system: f i r s t , there was a roya l ty tax per ton of copper produced at roughly 13.5 percent of the London Metal Ex-change (LME) monthly average pr i ce mines K16 per ton; then, there was an ex-port tax ( introduced in 1966) o f 40 percent of the LME p r i c e of K600 per ton ; in add i t i on , there was a corporat ion tax at 37.5 percent on the f i r s t 44 K200,000 of the net p r o f i t s and 45 percent on the r e s t . The e f f e c t s o f stagnating mine development were fu r ther re in fo rced by f a l l i n g ore grades and diminishing a c c e s s i b i l i t y o f the resource. In Nchanga mine, f o r i n -stance, the ore m i l l ed per ton of copper increased by 55 percent over the per iod 1956-69. Moreover, in Rhokana underground mine a cumulative deter -i o r a t i on o f working condit ions ( r i s i n g pressure and temperature with inc reas -- 32 -TABLE IV DEVELOPMENT PLANNING IN ZAMBIA: PERFORMANCE OF TDP AND 1 FNDP 1964 1965 1966 1967 1968 1969 1970 A GROSS DOMESTIC PRODUCT (Km) a 1 Planned (1964)prices) 457 511 570 637 711 795 888 2 Actual (1964 pr i ces ) 468 558 637 668 715 884 857 3 A c t u a l , growth rate .(%) - 19.2 14.1 4.9 7.4 23.2 -3.1 B GROSS FIXED INVESTMENT (Km) 1 Planned (1964 pr i ces ) 76 91 no 132 158 190 227 2 Actual (1964 pr i ce s ) 76 111 151 164 185 215 238 3 A c t u a l , growth rate (%) - 45.0 36.0 9.0 12.7 15.8 11.1 4 Actual as % of GDP 16.3 19.9 23.7 24.6 25.8 24.3 27.8 C PRIVATE CONSUMPTION (Km) 1 Planned (1964 pr i ces ) 203 228 257 288 324 364 405 2 Actual (1964 pr i ces ) 215 250 276 333 327 319 336 3 A c t u a l , growth rate (%) - 16.0 10.6 20.5 -1.7 -2.6 5.3 D EMPLOYMENT ('000) 1 A c t u a l , 'total 270 296 337 350 354 352 (390) b 2 A c t u a l , non-mining 217 244 282 295 299 285 (326) C E MINING CONTRIBUTION (%) * 1 Share in GDP 45.8 43.2 45.7 41.3 41.0 50.3 38.2 2 Share in Gov ' t Revenues 53.0 71.0 64.0 53.0 60.0 59.0 52.0 3 Share in Exports 92.0 93.0 95.0 94.0 96.0 97.0 97.0 4 Share in Employment 17.8 17.0 15.5 15.1 15.0 15.5 15.3 a.Km = m i l l i o n Kwachas; b. Planned 407; c. Planned 350 Sources: A,B, and C: Second National Development Plan (SNDP): Tables 1,2, and 3; D: SNDP: Table 9 and McPherson [1976b: Table I I .5]; El and E2: Table XI of Chapter 5 and Tables XLI, XLII and XLIII of Appendix C; and E3 and E4: Mining Yearbook of Zambia (1972 issue) - 33 -. 45 ing depths) was encountered. The message was well received by the Zambian government: major r e -forms o f the mineral resource management and economic development p o l i c i e s were c a l l e d f o r . F i r s t , there was a need fo r contro l o f the copper i n -dustry which would obtain f o r the government a large share of economic rent while encouraging mine development without ignoring the f i n i t enes s o f Copperbelt ' s l i f e . Then, there was a need f o r a system of a l l o c a t i n g of mineral revenues and c o n t r o l l i n g of the economy that would acce lera te c a p i -ta l accumulation without neg lect ing the need fo r improvement in the l i v i n g standards of a growing populat ion. The response of the Zambian government to th i s message was t rans la ted into the na t i ona l i z a t i on of 51 percent of the copper industry in 1970 and cons iderable c e n t r a l i z a t i o n of the economy at the same time. On August 1969 the Zambian government i n v i t e d the two fore ign mining companies, the Anglo-American Corporation (AAC) and the Roan Se lec t ion Trust (RST) to s e l l 51% of t h e i r shares to the government. There was an element of expropr ia t ion in th i s takeover to the extent that the government's pay-ments were to be made from the p r o f i t s accrued by the 51% shareholding. A l l mining, smelting and r e f i n i n g operations prev ious ly under RST and AAC were now consol idated into two s t a t e - c o n t r o l l e d companies, the Roan Consol idated Mines (RCM) and the Nchanga Consol idated Copper Mines (NCCM) re spec t i ve l y . The Mining Development Corporation (MINDECO) was then created to hold the 51% government i n t e r e s t in these two companies. The former owners, RST and AAC, in add i t ion to minor i ty shares, were to manage RCM and NCCM re spec t i ve l y under management and technica l consultancy contracts in order to "ensure con-t i n u i t y o f management and f a c i l i t a t e the retent ion of technica l management - 34 -and know-how, which the Zambian government could not a f fo rd at the time of 46 the takeover." Under the management contracts RST and AAC were given exc lus ive r i ght s in running the day-to-day mining operations inc lud ing the r i gh t to f i r e and h i re the. labor f o r ce . Management fees were s p e c i f i e d in terms of a percent-age of the p r o f i t s (2 percent a f t e r mineral tax but before income tax) plus 0.75 percent of to ta l proceeds in order to induce the fore ign managers to minimize production cos t s . NCCM and RCM were run as p r i va te en te rpr i se s . Capita l expenditure was t reated as operat ing cost and was f u l l y deduct ib le f o r taxat ion purposes. A mineral tax of 51 percent was imposed on the net operat ing p r o f i t (gross revenues minus operat ing cost minus cap i t a l expend-i t u r e ) and an income tax o f 45 percent on the remainder. Thus, the to ta l government revenues from mining operations amounted to 73.05 percent o f the . , * 46a net operat ing p r o f i t s . Output expansion and mine development were ' p o l i c y instruments ' of the government but the fore ign d i rec tor s have veto power over these dec i s ions . This became a source of c o n f l i c t which led to the termination AAC's and RST's management contracts in 1974 and 1975 re spec t i ve l y . The s t a te - con -t r o l l e d NCCM and RCM were made responsib le f o r r e c r u i t i n g t h e i r own managers and a new 100 percent state-owned company (MEMACO) was created f o r market-ing the Zambian copper. In a l l other respects the 1970 reforms continue to hold. The government acquired f u l l contro l of copper production and mine development p o l i c i e s , but the copper industry continued to r e l y qu i te heav-i l y on fore ign managerial and technica l personnel and substant ia l quant i t ies o f imported inputs f o r i t s operat ion. Furthermore, 49 percent of NCCM and 47 RCM continued to be owned by fore ign i n t e r e s t s . - 35 -The na t i ona l i z a t i on of the copper industry in 1970 was a part o f broader economic reforms which re su l ted i n cons iderable c e n t r a l i z a t i o n of the economy. In f a c t a takeover of 51 percent of the ownership shares of the manufacturing and importing indus t r ies preceded the takeover o f the mines. The Industr ia l Development Corporation (INDECO), a state-owned company, was created as ea r l y as 1966 to spur general i ndu s t r i a l development and by 1971 i t c on t ro l l ed more than 80 subs id iary and assoc iated companies. INDECO1s a c t i v i t i e s range from manufacturing and construct ion to serv ices and rura l en te rp r i se s . INDECO's net assets increased from K35.5 m i l l i o n in 1968 to K233 m i l l i o n in 1972, as a r e s u l t of the increased c e n t r a l i z a t i o n of 48 the economy a f t e r the 1970 reforms. In order to coordinate economic dev-elopment planning implemented through INDECO with mineral resource manage-ment exerc i sed through MINDECO, the government placed these two corporat ions under the overa l l superv is ion of new state holding corpora t ion , ZIMCO (Zambian Industr ia l and Mining Corporat ion) . Rural development p o l i c i e s were exerc i sed through the Zambian Rural Development Corporat ion. The Second National Development Plan (SNDP) covering the years 1972 through 1976 aimed at using these ' p a r a s t a t a l ' corporations to implement a po l i cy of " import subs t i tu t i on and progress ive reduct ion in the r e l a t i v e 49 share of the mining sector in the national output." The SNDP projected K2.161 m i l l i o n cap i t a l expenditure and envisaged a 15 percent annual growth rate of manufacturing, 6 percent fo r a g r i cu l tu re and 6.1 percent in mining. The GDP in 1969 pr ices was expected to increase from K1080.7 m i l l i o n in 1971 to Kl,505 in 1976 to enable an increase in rea l GDP per cap i ta from K246 in 1971 to K300 by 1976. Considerable a t tent ion was a lso given to the promot-ion of rura l development and non-subsistence employment. - 36 -In a mid-term review (1974) the Zambian planners admitted: "The r e -cent developments in the world copper market have once again come to cast a dark shadow on the country ' s economic f u t u r e . . . the SNDP targets in real terms are not expected to be r e a l i z e d and ' ca r ry over ' programmes ...may 50 turn out to be s i z e a b l e . " Although the energy c r i s i s of the ear l y 1970's and the c lo s ing of the Rhodesia-Zambia borders in 1973 may have caused SNDP to d e f l e c t from i t s course, the main source of i t s f a i l u r e i s to be foundd in the lack of long-run development planning. 2.5 The Need f o r a Long-run Development Plan Both FNDP and SNDP were based on short -run expectations of future cop-per p r i c e s , which in ne i ther case were r e a l i z e d . Project ing the depressed copper pr ices o f the preceding period 1967-65 (see Table X), FNDP was formu-l a ted on the pe s s im i s t i c p r i c e expectat ion of K600 per ton. As i t was, the very f i r s t year o f FNDP's implementation (1966) world copper pr ices jumped by 53 percent, from K502 to K769, and continued to r i s e over the en t i re 5 year planning hor izon, reaching an a l l - t i m e maximum of 996 in 1970 (by 60 percent higher than a n t i c i p a t e d ) . Zambia was unable to take f u l l advantage of these unusually high p r i c e s , as the necessary prov is ions f o r investment in maintaining and expanding mine capaci ty were not made (see Table XXXIV). This was so despite FNDP's ambitious output target o f 858,000 tons annually and the e x p l i c i t recogn i t ion o f the need fo r new mine development. Over-react ing to th i s lack of mine development the Second National Development Plan 1972-76, c a l l e d f o r a 40 percent expansion of mining cap-a c i t y by 1976. Seidman [1973:23] estimated that attainment of th i s target would requ i re mining investment at the rate o f K56 m i l l i o n annua l ly , " a l -most one t h i r d of the to ta l ex i s t i n g assets of INDEC0" (government's i n -- 37 -strument of i ndu s t r i a l development) and "almost-two times the annual govern-51 ment expenditure on rura l development." Such a massive mining investment was based on the over -op t im i s t i c p r i ce expectat ion of 800 c . i . f . (740 f . o .b . ) which, again, was not r e a l i z e d . The very f i r s t year of the Plan (1972) the copper p r i ce dropped to K690 per ton, 45 percent below i t s 1970 leve l and did not recover un t i l 1974. Obvious ly, short-run f l uc tua t ions -an inherent feature of the world cop-per market-have des t ruc t i ve e f f ec t s on short-run development planning. Zambia's experience in th i s respect , points to the need f o r : '(ii) concerted act ion with other copper exporters to s t a b i l i z e copper p r i c e s ; and ( i i ) pro-gress ive reduct ion of Zambia's dependence on copper thnough the formulat ion of resource management and economic development p o l i c i e s based on long-run pr i ce expectat ions. S t a b i l i z a t i o n of copper pr ices has already been attemp-ted through the formation of the Intergovernmental Council o f Copper Ex-52 port ing Countries (CIPEC) by Zambia, C h i l e , Z a i r e , and Peru in 1967. CIPEC, however, was not met with much success, due to i t s small share of the market and the r e l a t i v e l y high p r i ce e l a s t i c i t i e s o f world copper demand 53 and of non-CIPEC copper supply. Reduction of Zambia's dependence on copper, was the main ob jec t i ve of the f i v e - y e a r plans s ince independence. A comparison of Tables III and IV (rows E1-E4) reveals that t h i s ob jec t i ve was not achieved during the FNDP and, as the planners admitted, not much progress has been made during the 54 SNDP. Reduction of the economy's forty-year-il:ong dependence on copper r e -55 quires a long process of s t ruc tura l change which cannot be se r i ous l y pur-sued in the context of f i v e - y e a r plans. Short-term planning i s usual ly f a c -ed with the dilemma of e i t he r a l l o c a t i n g too much investment to mining (e.g. - 38 -SNDP) in which case the dependence on copper i s perpetuated, or a l l o c a t i n g too l i t t l e investment to mining (e.g. FNDP) in which case the source of revenues f o r non-mining investment is soon dr ied up. The optimal a l l o c a t i o n of revenues between mining and non-mining i n -vestment can only be determined.in the context of a long-term development 56 p lan , say over a twenty-year per iod. Such a plan should be based on: (a) long-run p r i ce expectat ions; (b) knowledge of the mining and non-mining pro-duction technolog ies ; and (c) information on the s i ze and q u a l i t y of the copper resource. Long-run p r i ce expectations would help to determine the approximate copper revenues f o r any chosen leve l of output oyer the planning hor izon, whi le the production technologies would determine the cost of pro-ducing that l eve l of output given input p r i c e s . In p a r t i c u l a r , the mining technology would determine the opportunity cost o f . these inputs . U l t imate ly , the f e a s i b i l i t y o f the plan would be determined by the s i ze and qua l i t y of + u 57 the copper resource. Formulation of such a long-term plan i s poss ib le (see Chapter 3 be-low). Knowledge of the production technologies can be acquired through est imation of appropriate production and/or cost funct ions using h i s t o r i c a l data on p r i c e s , inputs and outputs (see Chapter 4-7 below). S i m i l a r l y , i n -formation on the s i z e and qua l i t y of the copper resource may be obtained by analyzing h i s t o r i c a l data on past product ion, ore reserves, and average grade of the ore (see sect ion 5.5 below). F i n a l l y , long-run p r i ce expectations may be based on project ions and forecasts o f world copper demand and supply over the planning hor izon, with due cons iderat ion of the increas ing supply from s e r a p h and copper 's v u l n e r a b i l i t y to subs t i tu t i on from aluminum and p l a s t i c s (see, f o r ins tance, Takeuchi [1974], Pindyck [1976], Leont ie f [1977], and sect ion 8.2 below). - 39 -Formulation of a long-term development plan was also proposed by Seidman [1973] in a c r i t i c a l review of past resource and development p o l i c i e s in Zambia with p a r t i c u l a r emphasis on the SNDP: There i s another way to look at the government's s trategy of increas ing investment in copper; that i s , from the point of vriiew of opportun i t ies foregone to develop other sectors o f the econ-omy.. [T]he government should formulate a long-term i n d u s t r i a l and a g r i c u l t u r a l development strategy-say over a twenty-year p e r i o d . . . This approach does not argue that the copper mines should be neglected. On the contrary the fore ign exchange earn-ings and tax revenues which the mines earn can and should make an important cont r ibu t ion to the implementation of [ t h i s ] . . . a l t e r n a t i v e development s t rategy. This cont r ibut ion should be maximized with in the framework of a r e a l i s t i c ana lys i s of the world market...59 In the face of pe r s i s t i n g subsistence leve l s o f consumption, rap id ly growing popu lat ion, and gradual ly dep let ing mineral resource there i s pres -s ing need fo r such a long-term integrated resource management and develop-ment p o l i c y : ( i ) to improve the current standards of l i v i n g : and ( i i ) to provide an i ndu s t r i a l base as an a l t e r n a t i v e to the deplet ing resource. This need i s not s p e c i f i c to Zambia but to a l l mineraleexport ing develop-ing count r ie s . It i s true of the other copper exporters , Z a i r e , Peru, C h i l e , Botswana, Uganda and Papua New Guinea. The same holds f o r the t i n exporters , Malays ia, B o l i v i a , Tha i l and, Congo, and Burma; the bauxite ex-porter Jamaica and Guayana; the riiron exporters , L i b e r i a , Guinea and « Maur i tan ia, and others . - 40 -FOOTNOTES TO CHAPTER 2 1. See Tables III and IV. 2. See Seidman [1972:10. 3. McPherson [1976:7]. 4. For a more de ta i l ed h i s tory of Zambia ( inc lud ing p o l i t i c s and econ-omics) from ancient times to independence see Hall [1965]. For the more recent p o l i t i c a l h i s tory of Zambia as i t re la tes to the m u l t i -nat ional mining companies see Sk lar [1975]. 5. See Baldwin [1966:81], Martin [1972:30], and Drysdal l [1972:54-61]. 6. The c r u c i a l breakthrough was made in 1923 when a d r i l l - h o l e at Nkana in ter sec ted "c lean sulphide ore beneath a covering of oxides" (Drysda l l , 1972:60), but i t s s i gn i f i c ance was not r e a l i z e d unt i l i t reoccurred in 1926 at Roan Antelope. 7. Drysdal l [1972:58]. 8. Roan i s an abbrev iat ion fo r Roan Antelope copper mine(s). 9. Throughout th i s study "world market" i s def ined as to exclude the c e n t r a l l y planned economies. 10. See Baldwin [1966:3]. 11. K i s the abbreviated form of Kwacha, Zambia's nat ional currency s ince 1968. Pre-1968 s t a t i s t i c s were converted to Kwacha fo r comparabi l i ty at the rate one Zambian pound = 2 Kwachas = U.S. $2.80; Kmn = m i l l i o n Kwachas. 12. Of course, what i s of importance to the country i s the behaviour of copper pr ices r e l a t i v e to the behaviour of import p r i c e s . The l a t t e r had a smooth slow upward trend (see McPherson [1976b] f o r the uni t value indexes of exports and imports) . 13. For ins tance, fo r 1955-58 and 1970-72 output and exports compared as fo l lows: 1955 1T956 1957 1958 1970 1971 1972 Output ('000 metr ic tons) 359 404 436 400 684 651 717 Exports ('000 metr ic tons) 348 374 415 409 684 635 711 Source: f o r output, Table X; fo r exports , McPherson [1966:D.2.13]. 14. See Table XV, column QQCU. 15. Baldwin [1966:16-17]. 16. Ib id: 29 - 41 -17. Ib id. 18. Coleman [1971:146]. 19. See Baldwin [1966:30]. 20. Throughout the present study the terms "mining sector " and "copper i n -dustry" are used interchangeably. Z inc , lead , coba l t , and coal are a l so produced but are not taken e x p l i c i t l y into account f o r 3 reasons: ( i ) There are no comprehensise separate data on the non-copper mining subsector; ( i i ) some of these minerals are produced as by-products o f the copper indus t ry ; and ( i i i ) copper accounts f o r approximately 95 per-cent o f the to ta l value o f mineral product ion. 20a. By 1964 the cont r ibut ion of Manufacturing to GDP at f ac to r cost was K28.2 m i l l i o n (5.4 percent o f to ta l GDP) o f Transport and Communi-cat ions K20.6 m i l l i o n (3.9 percent ) , and of construct ion 20.0 m i l l i o n (3.8) percent (See Tables XLI, and XXXIX o f Appendix C) . 21. Baldwin [1966:40-41]. 22. Martin [1972:38]. 23. Ib id. 24. Martin [1972:36-37]. 25. Report of the UN./E.C.A./F.A.0. Economic Survey Mission on the Economic Development o f Zambia, 1964, headed by Professor Dudley Seers. Quote obtained i n d i r e c t l y from Martin [1972:38]. 26. The share o f Manufacturing and Agr i cu l tu re in GDP are ca l cu la ted on the basis of data given in Tables XLI and XXXIX. The data on education are obtained from Fry and Harvey [1974:196], 27. See Martin [1972:36]. 28. See Tables XIV and XV. 29. Seidman [1972:9]. 30. Legum [1966:60]. 31. Mart in [1972:46]. 32. See footnote 25, th i s Chapter. 33. "Wage employment" and "non-subsistence employment" are used i n t e r -changeably. 34. F i r s t National Development Plan 1966-70 [1966:31]. - 42 -35. Ib id. :32. 35a. Note that the GDP d e f l a t o r does not include copper export p r i c e s . 36. Second National Development Plan 1972-76 [1971:3]. 37. 1 Zambianizat ion ' i s government's term for the po l i cy of rep lac ing expatr iates by Zambian na t iona l s . In 1966 the M in i s ter o f Labor set up a Zambianization committee f o r the copper industry . 38. Second National Development Plan 1972-76 [1971:1]. 39. Ib id . :4 . 40. There i s some discrepancy among the production s t a t i s t i c s from d i f f e r -ent sources e s p e c i a l l y f o r the years 1965, 1969, and 1970. Drysdal l (1971 gives 685.1 metric tons fo r 1965 and 754.2 tons f o r 1969. McPherson [1966] and the Zambia Mining Yearbook give 695.7 tons fo r 1965, 719.5 tons f o r 1969 and 684.1 tons fo r 1970. The SNDP [1971] gives 675.0 tons fo r 1965 and 684.7 tons f o r 1970. 41. See Drysdal l [1972:73]. 42. Drysdal l [1972:72] reports that although the Bancroft mine of the M G group increased i t s production by 35 percent, th i s was more than o f f -set by the f a l l of production in Rhokana and Nchanga mines due to f a l -l i n g grades and diminishing a c c e s s i b i l i t y of the ore (see below). 43. See Simwinga [1975:345] and Drysdal l [1972:73]. At his famous Mulun-gushi speech President Kaunda declared "I want to say to the mining companies that I am very disappointed at the v i r t u a l lack of mining development s ince independence". Kaunda, K. [1968]; Zambia Economic  Revolut ion, Lusaka 215 44. For a de ta i l ed desc r ip t i on of the taxat ion system during the per iod 1964-1969 see Harvey [1971] and Harvey [1972:132-35]. 45. Drysdal l [1972:72]. 46. Simwinga [1975:87]. 46a. Before compensation payments to the companies. 47. A number of studies were wr i t ten on the legal and economic aspects of the Na t i ona l i za t i on of the Zambian copper industry . Of p a r t i c u l a r inQ te re s t are the studies included in Faber and Potter [1971], Bostock and Harvey [1972] and Seidman [1975]. 48. Seidman [1975:369]. 49. The Object ives arid Strategy of the Th i rd National Development Plan ]1976:1], Annexure. - 43 -50. Ibid :2. 51. Seidman [1972:23]. 52. The object ives o f CIPEC are, in genera l , to promote the cont r ibut ion of copper resources to the economic development of i t s members and, in p a r t i c u l a r to "coordinate measures designed to f o s te r real earnings from copper expor te r s . . . [and] assure greater p r i ce s t a b i l i t y on the i n -ternat iona l copper market". De V l e t t e r [1972:255]. 53. In 1969 the approximate share of CIPEC: ( i ) in world copper production was 33.0 percent; ( i i ) in world copper reserves was 48 percent; and ( i i i ) in world copper exports was 60 percent. (See Takeuchi [1972]). Ava i l ab le estimates of the p r i ce e l a s t i c i t y of world copper demand, w, range between 0.1 and 0.3 in the short -run (See Takeuchi [1971] and F i sher , Cootner, and Ba i ly [1972])and around 0.8 in the long-run (See Pindyck [1976]). We are less ce r t a in on the order of magnitude of the p r i ce e l a s t i c i t y o f non-CIPEC copper supply, n, s ince i t depends on the cost range and CIPEC 1s share cons idered. Takeuchi [1972] i n fe r s an e l a s t i c i t y in the range of 0.16-0.30 in the short-run and 0.7 in the long-run while Pindyck [1976] reports 0.20 and 1.6 r e spec t i ve l y . On the basis o f the above f igures we may ca l cu l a te the p r i ce e l a s t i c i t y of demand for CIPEC copper as: E = (w/s) - (n/s) + n. (For a der i va t ion of th i s formula see Takeuchi [1972]. Only i f E<1 CIPEC could be success-fu l in maintaining r e l a t i v e l y high copper pr ices by fo l lowing an 0PEC-type s t rategy. Both Takeuchi ' s c a l cu l a t i on s based on e a r l i e r estimates of w and n and our ca l cu l a t i on s based on more recent estimates point to an E > 1 except f o r the very short -run and provided CIPEC increase i t s market share by opening i t s doors to other copper-export ing deve l -oping countr ies (e.g. P h i l i p p i n e s , Papua New Guinea, Botswana, and o ther s ) . There i s fu r ther the p o s s i b i l i t y o f c reat ing a buf fer stock but past experience with other commodities points to severe d i f f i c u l t i -es of f inanc ing and administer ing such a stock (see Brown and Bu t t l e r [1968]). 54. In a mid-term review of the SNDP the planners s ta ted : "the inherent weakness of the economy remains, depending as i t does on the f l u c t u a -t ions in the world, p r i ce of copper". The Object ives and Strategy of the  Th i rd National Development P lan, Annexure, p. 2. 55. S t ructura l change i s used here, in i t s conventional sense in the deve l - •:• opment l i t e r a t u r e , i . e . the gradual s h i f t from primary production to secondary i ndu s t r i e s . In our context, primary production re fer s to mining and secondary indus t r ie s to manufacturing and commerical a g r i -c u l t u r e . 56. For a good treatment of the technica l aspects o f short-term versus long-term economic planning see Heal [1973]. 57. Of course, long-run p r i ce expectations may not be r e a l i z e d and the s i ze of the resource under-or over-est imated. In a world of l im i ted fo res i gh t and imperfect knowledges r e a l i s t i c plan must involve a - 44 -continual planning rev i s i on as demonstrated by Goldman [1968]. 58. The p o s s i b i l i t y o f copper recovery from unconventional sources such as ocean- f loor nodules should a lso be cons idered. 59. Seidman [1973:32-33]. CHAPTER 3 MODELLING OPTIMAL MINERAL DEPLETION IN DEVELOPING ECONOMIES: THE ROLE OF PRODUCTION TECHNOLOGIES The purpose of th i s Chapter i s t h ree fo l d : f i r s t to develop an i n t e r -temporal model o f optimal cap i t a l accumulation and mineral resource deple^ tion in the context o f a developing economy; second to derive some q u a l i t a -t i v e e resu l t s regarding the optimal rate of ext rac t ion and a l l o c a t i o n of mineral revenues among a l t e r n a t i v e uses, and to provide an economic i n t e r -pretat ion of these r e s u l t s ; t h i r d , to demonstrate the ro le of production technologies in the formulat ion and optimal so lu t ion of the model. The funct iona l s p e c i f i c a t i o n and empir ica l est imation of the product-ion technologies i s perhaps the most u n s a t i s f a c t o r i l y dea l t with feature of ex i s t i n g planning models. To quote Sims and Gotsch [1971]: "Perhaps the most ser ious l i m i t a t i o n to improving planning methods has been the r e l a t i v e neglect of t h e i r empir ica l foundat ions. . .Even though very simple formulations of econ-omic re l a t i on s may have been j u s t i f i e d 10 years ago by the lack of time s e r i e s , i t i s no longer necessary to descr ibe the economic s t ructure in such s i m p l i f i e d terms.. . [ P l a n -ners have in general been slower than econometricians to experiment with a va r ie ty of funct iona l forms and with v a r i -ables (such as pr i ces ) which may enter behavioral r e l a t i o n -ships in computational ly inconvenient waysJ The conventional p r ac t i ce i s to spec i fy simple funct iona l forms such as the Leont ie f , Cobb-Douglas and CES production funct ions (see f o r instance Bruno [1966], Kendrick and Tay lor [1971] and Chenery and Raduchel [1971]). These funct iona l forms, however, are known to impose a p r i o r i r e s t r i c t i o n s on the e l a s t i c i t i e s o f f a c to r s u b s t i t u t i o n , which have ser ious impl icat ions f o r i n -tertemporal op t ima l i t y . Thus, in formulat ing our intertemporal model p a r t i -cu l a r emphasis w i l l be placed on production technolog ies. Moreover, the - 46 -impl i ca t ions f o r opt ima l i ty of a p r i o r i r e s t r i c t i o n s on these technologies w i l l be i nves t i ga ted , by expressing the opt ima l i ty condit ions in terms of e l a s t i c i t i e s and s t a t i c shadow pr ices der ived from these technolog ies. The Chapter i s organized into e ight sec t ions . Sections 3.1 and 3.2 discuss the general assumptions and the formulat ion of the planning problem. Section;. 3.3 introduces the ob ject i ve funct ion and the terminal cond i t ions . Section 3.4 the i n i t i a l resource condit ions and dynamic con s t r a i n t s , and sect ion 3.5 the production technolog ies. The complete model, the opt ima l i ty condit ions and t h e i r interpretat ionrtare presented in sect ion 3.6. In sec-t ion 3.7 the ro le of production technologies i s demonstrated by expressing the opt ima l i t y condit ions in terms of e l a s t i c i t i e s and s t a t i c shadow pr ices derived from these technolog ies. The C U-,J"> i ;U iv.*.~.*• s . . . 3.1 Assumptions As the emphasis i s on production technolog ies , several complicat ions such as the investment in human c a p i t a l , the export and import of cap i t a l and balance of payments problems have been assumed away. This i s not to de-emphasize t h e i r importance in r e a l i t y but to keep the model with in manage-able proport ions. The assumptions are numerous: ( i ) GNP per cap i ta is a sound i nd i ca to r o f economic development. 1 No interpersonal income-d i s t r ibut ion problems are cons idered. (Expatr iate mining labor i s not considered part of the popu-l a t i o n ; nor are i t s earnings part of GNP). ( i i ) There i s a Central Planning Author i ty , t o ta l l y , g in charge of the country ' s development and natural resource p o l i c i e s , ( i i i ) The mining industry is not an integra l part of the economy in the fo l lowing sense: i t i s operated by a fore ign company, - 47. -i t employs substant ia l quant i t ies of fore ign inputs , and produces an output which i s not consumed domest ica l ly (see assumption ( iv ) below). However, a l l long-run p o l i c i e s a f f e c t i n g the mining industry ( inc lud ing the rate of pro-duction) are made by the Planning Author i ty . The fore ign company simply minimizes the operat ing costs o f producing a given leve l output Q preassigned by the Planning Author-i t y . 2 The en t i r e mineral output i s exported at i t s world p r i c e . A part o f the mineral revenue i s used fo r the payment of the serv ices o f fore ign inputs while the res t i s used in importing a homogeneous commodity Y, which can e i t h e r be consumed or invested. The country i s small enough f o r i t s t rad ing a c t i v i t i e s not to a f f e c t the pr ices of i t s exports and impor t s . 3 The non-mining sector employs domestic labor and cap i ta l (accumulated Y) to produce add i t iona l quant i t i e s o f the homogeneous commodity Y f o r purely domestic use. Although the en t i r e populat ion pa r t i c i pa te s in consumption, the labor force cons is ts of only the port ion of the popu-l a t i o n engaged in market-or iented a c t i v i t i e s . Subsistence a g r i c u l t u r e , where the bulk of the populat ion i s usual ly 4 engaged, does not enter the production s ide of the model. Thus, the domestic production of Y i s net of subsistence output. The labor f o r c e , as def ined above, i s a f i xed pro-port ion of the populat ion. This assumption can be e a s i l y - 48 -relaxed to incorporate either an exogenous mechanism of rural-urban migration or an endogenous mechanism of trans-forming subsistence workers into industrial labor through 5 investment in human cap i ta l . ' (v i i ) The country neither exports nor imports capita l . In con-trast to the tendency of oil-producing countries (which have small absorptive capacity domestically) to invest in foreign assets, the mineral-exporting countries have in -deed concentrated their efforts on domestic development, ( v i i i ) There is a balance of payments equilibrium, at a l l times, in the sense that the value of exports equals the sum of the value of imports plus payments to foreign inputs. (There is no servicing of foreign debt).^ A fixed exchange rate is assumed.^9 (ix) Conditions of certainty prevai l . In particular, the size and the quality of the mineral resource is known with cer-tainty. We are concerned with a "mature" mineral exporting country, where a comprehensive exploration of i ts ore bodies has already been undertaken.^ (x) F inal ly, i t is assumed that while the resource is hetero-geneous, i t can be separated into chunks of separate grades g so that best grades can be (and are) mined f i r s t . Assumptions (i) through (v) while based on the Zambian case are not total ly unrealistic for most mineral-exporting developing countries. The purpose of these assumptions is to set the stage for the rest of the model. Assumptions (vi) through (x), on the other hand, are made in order to keep - 49 -the model manageable. The imp l i ca t ion o f assumption (v i ) i s that i t de-priwes "development" of part o f i t s conventional meaning as the process through which the subsistence sector i s modernized.. Assumptions ( v i i ) and ( v i i i ) are strong assumptions but easy to re lax i f we are prepared to i n -troduce more e x p l i c i t l y the fore ign sec tor . These assumptions imply that e i the r the return on fore ign assets (or the cost of external borrowing) i s p r o h i b i t i v e l y low (high) or the country exh ib i t s some avers ion towards i n -creas ing i t s 'openness 1 . Assumption ( ix) serves to keep the model determ-i n i s t i c and to avoid the complicat ions that uncerta inty might have i n t r o -duced. (For a s tochas t i c exp lorat ion model see Uhler [1973]). To the extent that the mineral resource i s not known with ce r ta in ty s e n s i t i v i t y ana lys i s may be employed to study the impl icat ions of a l t e r n a t i v e i n i t i a l resource stock s p e c i f i c a t i o n s . F i n a l l y assumption (x) i s admittedly r e s t r i c t i v e though!" not u n r e a l i s t i c at l eas t in the case of copper in Zambia; other l o -cat ions and other minerals may not f i t th i s pattern in which case the problem of optimal ' b lend ing ' o f d i f f e r e n t ore grades must be reso lved. Since th i s problem has been barely touched in the received l i t e r a t u r e (see He l l iwe l l [1977]) we make only a modest attempt of incorporat ing resource heterogen-e i t y into the model by cons ider ing only the case in which best grades can be mined f i r s t . Other assumptions concerning the nature of s p e c i f i c t heo re t i c a l r e -l a t i ons and the behaviour of p a r t i c u l a r economic agents w i l l be made in the course of the presentat ion. 3.2 Formulation of the Planning Problem Consider a developing economy with a r i c h mineral endowment and two d i s t i n c t production technolog ies: A mining technology which combines f o r e i -= 50 -gn inputs, M, and domestic labor, L, with mining structures, Z, and the mineral resource, R, to produce refined metal, Q, which is transformed into a homogeneous commodity, Y, through international trade; and, a non-mining technology that combines capital, K, with domestic labor, N, to produce add-itional quantities of Y for purely domestic use. Then, the total quantity of Y is: Y = Y1 + Y2 (3.1) where Y-j is the quantity of Y produced domestically and Y2 is the quantity of Y imported in exchange for the exported Q. The twp ways of acquiring Y are described by: Y1 = *.(K,N) (3.2) Y2 = O(Q) = 9[F(M,L;Z,R)] (3.3) where $(•) is the non-mining technology, F(«) is the mining technology and Q ©(•) is a foreign trade transformation function converting units of Q into 9a units of Y, after deduction of payments to foreign inputs. The homogeneous commodity Y/ can be either consumed or invested. Act-ually, there are three distinct uses of Y: (i) i t can be consumed to raise society's standard of living; (ii) i t can be invested in mining structures such as mine shafts and mills to maintain or increase the production of Q which can be exchanged abroad for more Y; and ( i i i ) i t can be invested in non-mining capital to maintain or increase the domestic production of Y. Assuming Y is exhausted between these three uses we may write: Y = X + V + S (3.4) where X is consumption, V is mining gross investment and S non-mining gross investment. The optimal division of Y between X, V and S is an intertempo-ral ' allocation problem since current investment (and consumption) decisions - 51 -a f f e c t fu ture production and consumption. A second key intertemporal a l l o c a t i o n problem re la tes to the se l ec t i on of the optimal rate of e x t r a c t i o n , Q, which determines the a l l o c a t i o n of the f i xed mineral resource, R, between the present and the fu tu re . A higher Q today impl ies a resource stock f o r the future of smal ler quant i ty and i n f e r -i o r q u a l i t y , given assumption (x) above. There is a t h i r d a l l o c a t i o n prob-lem:!: the d i v i s i o n of domestic labor between the mining and non-mining sec -t o r s . This i s not an intertemporal problem, but i t i s e s sent i a l that the labor force B (which bears some f i xed re l a t i on sh ip to populat ion E) i s o p t i -mally a l l o ca ted between the two sectors . Assuming that a l l labor is employ-e d - - f. ..r.: B = 5-E = L + N (3.5) where £ i s a f i xed propor t ion , L i s mining labor and N non-mining labor. Imagine now that the voters of the country concerned e l e c t ifiof.a term o f T years a Central Government which promises to so lve these three a l l o -cat ion problems by choosing X, V, S, Q, L, N in such a way as to maximize 9b the fo l lowing soc ia l welfare funct ion : W (X , E) = U(X t , E t ) e " p t dt (3.6) where X^ i s consumption at time t| E^ i s populat ion at time t and U(-) i s a well-behaved u t i l i t y funct ion which i s discussed below. Parameters p and T, represent a f i xed po s i t i ve discount rate and a given f i n i t e planning h o r i -zon. The government planners need only choose, X, V, Q and L, s ince S and N are obtained as res idua l s from Y and B r e spec t i ve l y . The planners, however, are not completely free to choose values f o r the contro l s X, V, Q and L. They a re , f i r s t , constrained by the ex i s t i n g technologies $(•) and F(«) of producing Y and Q respec t i ve ly and the terms - 52 -of trade embodied in G(«) through which Q i s transformed into Y (see equa-tions 3.2 and 3.3). The planners' choice of controls i s further constrained by the i n i t i a l stocks of the inputs, K, Z and R, inherited from the past and by the size of.the population E. The i n i t i a l resource stocks determine how much can be produced i n the f i r s t year of the plan. The planner's objecr-tive i s to adjust these stocks over the planning horizon T by appropriate choice of the controls in order to maximize the social welfare function (3.6). Again, there are l i m i t s to the accumulation or decumulation of these stocks over the planning horizon. Terminal stocks that are too low may be 'unfair' to post-plan generations, and those that are too high, i n f e a s i b l e . To recapitulate, the planner's objective i s to maximize the objective function expressed by the social welfare function (3.6) subject to three groups of constraints: ( i ) the i n i t i a l resource conditions KQ, Z Q and RQ inherited from the past; ( i i ) the production technologies F(«) and $(•) and the foreign trade transformation function e(«h and ( i i i ) the terminal con-ditions that some prescribed resource stocks, K j , lj, and R-p, should be l e f t at the end of the plan. In the following three sections we discuss in more detail the objective function and the constraints. 3.3 Objective Function and Terminal Conditions The objective function of a planning model represents preferences a-9b mong alternative social states. In the context of highly aggregative long-term planning where there i s a single commodity, Y, which can be either con-sumed or invested, the objective function represents the structure of the intertemporal consumption preferences (see assumption ( i ) below). The social ordering of consumption sequences over time may be repre-sented by a social welfare function such as (3.6) which postulates that the - 53 -total uti l ity over the planning horizon.is given by the integral of dis-counted util it ies enjoyed at each instant in time. Welfare function (3.6) , further indicates that the planners "are not indifferent between a situation where a one man community has a high standard of living, and a situation where all members of a large community enjoy this standard." 1 0 If i t can be assumed that population growth is exogenously given, the social welfare func-tion (3.6) may be written 1 0 a as: J = / J u(x t) e ' p t dt (3.7) where x t = X t /E t is per capita consumption at time t i and u(«) is a well-behaved uti l ity function in the sense that is strictly concave and monoton-•itallyy increasing in x^« In particular u v = au/ax > 0 (3.8) X -u v v = 92u/8xl < 0 (3.9) Jiimuu = 0, and (3.10) A A X » Aim u = <» Thus, while additional consumption adds more satisfaction (3.8) , units of satisfaction are not equivalent Or proportional to units of consumption because of the assumed diminishing marginal util ity (3.9). The satiation point (u = 0) is approached only asymptotically as per capita consumption X tends to infinity (3.10). Property.(3.11), on the other hand, indicates that as per capita consumption tends to zero, marginal uti l ity becomes in-finity: (i.e. starvation should be avoided whateverr the opportunity cost.) - 54 -This ensures that any reasonable consumption path should avoid x^=o fo r any t . The funct iona l (3.7) provides a rule, by which the planners can attach numbers to d i f f e r e n t consumption paths and compare a l t e r n a t i v e consumption programs ava i l ab le to the soc ie ty over the planning hor izon. The e n t i r e future i s broken down into two par t s : the planning horizon of the f i r s t T years , and the i n f i n i t e future beyond T. The p lanners ' concern fo r the welfare of generations l i v i n g beyond T i s expressed in the terminal con-d i t i ons of the stock va r i ab le s : k(T). = k T (3.12) z(T) = z T (3.13) r (T) = r T (3.14) where k = K/E i s the non-mining cap i ta l stock per c a p i t a , z = Z/E i s the mining cap i t a l stock ( s t ructures ) per c ap i t a , and r = R/E i s the.resource stock per c a p i t a . Choosing the terminal stocks involves t r ad ing -o f f consumption with in the planning horizon fo r consumption a f t e r i t . C lea r l y th i s choice i s a r b i t -rary unless the planner knows ( i ) how long a f t e r T the world w i l l l a s t ( i i ) what post-plan consumption p o s s i b i l i t i e s are impl ied by the s p e c i f i e d t e r -minal stocks and ( i i i ) the form of the past-plan consumption preferences. While in the absence of such knowledge the choice of the terminal time and terminal stocks i s a p o l i t i c a l (or e t h i c a l ) d e c i s i o n , r e l a t i v e i n s e n s i t i v i t y of the plan to the terminal condit ions must be ensured. In theo re t i c a l models the planning horizon may be made i n f i n i t e to a-void the d i f f i c u l t i e s with spec i fy ing the end condit ions of the p lan. In - 55 -e m p i r i c a l l y or iented models, on the other hand, the planning horizon should be made as long as poss ib le subject to cost cons iderat ions and the dec l i n ing c r e d i b i l i t y o f data fo recas t s . . 1 1 Once the time horizon i s chosen "the most re levant question about the terminal cap i t a l stock s p e c i f i c a t i o n s i s the ex-tent to which the i n i t i a l phases of the plan are s en s i t i ve to them, s ince in actual planning app l i ca t ions "an optimal growth exerc i se would be rerun with new data every time a rev i sed perspect ive plan was f o r m u l a t e d " . 1 1 3 Thus, once some terminal stocks are s p e c i f i e d and an ' opt ima l ' so lu t ion is ob ta in -ed the s e n s i t i v i t y o f the i n i t i a l phases of th i s so lu t i on to changes in the terminal condit ions must be tes ted. S e n s i t i v i t y ana lys i s with respect to the length of the planning horizon may a lso be attempted. 3.4 I n i t i a l Resource Condit ions and Dynamic Constraints The i n i t i a l resource condit ions r e f e r to the quant i t ie s o f inputs i n -her i ted from the past which are essent ia l to the operat ion of the mining and non-mining production processes. They include the stocks of populat ion, mineral resources, mining s t ructures and non-mining cap i ta l at time t = 0. The dynamic constra ints express the time rate of change (A = dA/ dt) of these stocks due to the act ions of the planners and the mere possage of time. Denote the population l i v i n g at time t = 0 , by the equation: E (0) = E Q (3.15) E i s assumed to grow over time at the natural rate n, determined by b i o l o g -3bail 1 factors as f o l 1 ows: E. = E eP-t or E = n-E (3.16) L 0 j The labor f o r ce , B, i s assumed to be proport ional to the popu lat ion; thus, i t grows at the same rate n: B t = i • E t = B B e n t (3.17) - 56 -where 5 is a fixed proportion. The mineral resource stock at time zero, R(0), is assumed ta be of given quantity RQ. The stock, RQ, is the amount of given mineral contained 12 in ore deposits with a 'cut-off grade' higher than the metal content of common r o c k J 2 a Dividing R by E we obtain the mineral resource per capita as r = R/E. Then, the resource per capita at time zero is given by: r(0) = r Q (3.18) The country's initial mineral stock is non-replenishable by nature and thus, decumulates at the rate -Q where Q is the rate of extraction. In per capita terms the rate of change of the mineral stock is as follows: r = - Q / E - n r = - q - n r (3.19) where q = Q / E Eventually the mineral resource stock RQ may be exhausted and since a nega-tive resource stock is meaningless we should impose a non-negativity con-straint on the remaining stock at each point in time: Rt > 0 (3.20) Constraint (3.20) ensures that cumulative extraction over the time horizon T cannot exceed the total resource availability: / J Q t d t < R Q (3.21) In "mature" mineral-exporting newly independent countries, a substant-ial stock of mining structures such as shafts and mills has been put in place by foreign companies or colonial governments. Upon nationalization or independence, control over these structures passes to the national gov-13 ernment. Denote the stock of mining structures at time t = 0 by ZQ or in per capita terms: - 57 -z (0) = z Q h Z 0 / E 0 (3.22) As new investment in mining s t ructures is taking p l ace , z accumulates over time at the ra te : z = v - (6 2 + n ) z (3.23) where v i s per cap i ta gross investment in mining s t ruc tu re s , 6 2 i s a given constant deprec ia t ion rate f o r s t ructures and n i s the rate of population growth. Replacement investment equal to (6 2 + n) i s needed to counteract the forces of deprec iat ion and population growth'and to maintain the same r a t i o of s t ructures to populat ion. F i n a l l y the economy, however backward, i s l i k e l y to have i nhe r i t ed some stock of non-mining c a p i t a l , - K , or in per cap i ta terms, k (0) = k Q = KQ / E Q (3.24) where k is the cap i t a l /popu la t i on r a t i o . To the extent that economic de-velopment involves net investment in non-mining c a p i t a l , k w i l l be growing at the ra te : k = s - (6 1 + n ) k (3.25) where s i s gross investment per cap i ta and (6-j + n ) k replacement i nve s t -ment f o r maintaining the same cap i t a l per cap i ta as deprec iat ion and popu-l a t i on growth take place over timeJ3a We turn now to the technolog ica l constra ints or production processes which re l a te the above inputs to the produced outputs. 3.5 Technological Constra ints : Mining and Non-mining Production Technologies Production technologies are of p a r t i c u l a r importance f o r a number of reasons. F i r s t , they d i c t a te how much can be produced at each point in - 58 -time, given the init ial stocks of the inputs and their accumulation or de-pletion to that time. Second, they provide the marginal productivities or static shadow prices of the inputs on which their dynamic shadow prices and further accumulation or decumulation rest (see Section 3.7). Furthermore, the production technologies determine the rate at which one input may be sub-stituted for another in the production of given output as relative input prices change over time. Let us f i rst , consider the non-mining technology $(•)> represented by the production function (3.2). By assuming constant returns to scale we may rewrite (3.2) in per capita terms as: Y ] E Y] / E = * ( K/E, N/E) = <}> (k, h) (3.26) Where E / E is the per capita quantity of Y produced domestically and k E K / E and h = N / E are the per capita quantities of non-mining capital and labor respectively. The technology <f> (•) is assumed to be a non-negative, concave and increasing function of the input levels. In parti-cular, • (0) = 0 , <}>k > 0, and * n > 0 (3.27) * k k < 0 , * h h < 0 , and <)>kh > 0 (3.28)} Conditions (3.27) and (3.28) require positive but diminishing marginal pro-ducts for the two inputs. We now turn to the mriining technology. Equation (3.3) involves both a production function for Q, denoted by F(«)» and a foreign trade transfor-mation function e(«) converting Q into Y. We, f irst, discuss F(«) and then 0 (•)• Rewriting F (•) for convenience we have: Q = -F (M,' LV Z, R) (3,29) - 59 -14a where Q i s the indus t ry ' s output of re f ined metal , M i s the aggregate quant i ty o f a l l imported inputs , and L i s domestic l abor ; Z represents min-ing s t ruc tu re s , and R the mineral resource stock. While the planners in developing countr ies are usual ly involved in the choice of the rate of ex-t r a c t i o n (and r e f i n i n g ) , Q, and the leve l of investment in mining s t r u c t - , ures, they are not d i r e c t l y involved in the choice of M and L. The tend-ency i s f o r governments to na t i ona l i ze the mining i n d u s t r i e s , reserv ing f o r themselves a l l long-run" d e c i s i o n s ( i . e . decis ions a f f e c t i n g stocks) whi le a l lowing one or more fore ign companies to conduct the mining operations and 15 make a l l the short -run dec i s i ons , ( i . e . dec is ions a f f e c t i n g f lows) . Given the world p r i c e , P^, o f imported inputs , M, and a government-f i xed wage r a te , P^, f o r domestic l abor , L, the fore ign company chooses the flows of M and L to minimize the operat ing cost of producing given output Q with given f i xed inputs Z and R; i . e . , the f i rm solves the fo l lowing cost minimization problem: minimize {PM M + P,_ L : F(M, L; Z, R) > Q } = C (P M , P L ; Z, R, Q) (3.30) M, L where C denotes operat ing or va r i ab le cos t . Henceforth we r e f e r to M and L as va r i ab le inputs and to Z and R as f i xed inputs , ( f i xed from the point of view of the fore ign company). Correspondingly we r e f e r to C (« ) as the v a r i -able cost func t i on . E a r l i e r , in sect ion 3.2, we re fe r red to the problem of a l l o c a t i n g -labor between the mining industry and the res t of the economy. Here, we postulate that th i s a l l o c a t i o n is con t ro l l ed by the government in an i n -d i r e c t way by f i x i n g the mining wage rate and l e t t i n g the mining company em-ploy as much domestic labor as i t wants whi le the remainder i s employed in - 60 -the rest of the economy. Considering P L e x p l i c i t l y allows us to i n v e s t i -gate with in the model the condit ions under which domestic labor might i n -crease i t s share in the to ta l input payments v i s - a - v i s the share of fore ign inputs. Although th i s i s not an intertemporal problem, i t i s nevertheless important. In several developing countr ies the s e n s i t i v i t y o f c i t i z e n s t o -wards fore ign input payments in view of the s i ze of the mining industry has compelled governments to e i t he r adopt an e x p l i c i t mining wage po l i cy or to 15a support miners" union demands. Thus, we have two l eve l s of opt imizat ion : ( i ) the intertemporal (or long run) opt imizat ion by the countrys"s planner through the choice X,V,Q, and and ( i i ) the s t a t i c (or short-run) opt imizat ion by the company's man-agers through the choice of M and L. For tunate ly , the l a t t e r need not be considered e x p l i c i t l y . We know from dua l i t y t h e o r y ^ that i f fArms m in i -mize co s t , C (•)- contains the technolog ica l and economic information in the production funct ion and the cost minimizing behaviour (LHS o f equation 3.30). Assuming constant returns, to var i ab le inputs and d i v i d i ng by Q the RHS o f (3.30) becomes the unit va r i ab le cost func t ion : c = c (P M , P L ; Z, R) (3.31) where c = PjVj(M/Q)+P|_(L/Q) i s the operat ing or var iab le cost of producing one unit o f Q and c ( - ) i s a well-behaved unit va r i ab le cost funct ion : monotonic-a l l y i n c rea s i ng , concave and l i n e a r l y homogeneous in var i ab le input p r i c e s , P^, and P^, and convex in f i xed input quant i t ie s Z and R. For a more d e t a i l -ed d i scuss ion of the va r i ab le cost funct ion see sect ion 4.1. Here we are merely concerned with i t s general p roper t ie s . I f (3.31) i s well-behaved we may use Shephard's (1953) lemma to der ive the uni t demand funct ions f o r the var i ab le inputs . They simply co inc ide with - 61 -the partial derivatives of the unit cost function with respect to the vari-able input prices: m* =- 3c/9PM = fri(PM, P L; Z, R) (3.32) l* = ac/3PL = A (P M , P L; Z, R) (3.33) where m* and I* are respectively the cost-minimizing quantities of imported inputs and domestic labor per unit of output. To economize on notation and gain in clarity of exposition during the presentation of the optimal control model in the next section, we re-define m*=m, l*=l Pjv| = 9 and P^  = w. (We revert back to the original notation in Chapter 4.) Rewriting (3.31), (3.32) and (3.33) with the new notation we have: c = c (g, w; Z, R) (3.34) m = m (g, w: Z, R) (3.35) l = l (g, w: Z, R) (3.36) The following properties on (3.34) - (3.36) hold provided c(-) is a well be-haved cost function (Note that f = 9f/8x) X m = Cg > 0 and I = c w > 0 (3.37) m ? crtft••< 0 and a = c < 0 (3.39) • g • gg w ww • m• = c > 0 and I = c > 0 (3.40) w gw g wg ' The special nature of mining (see section 4.1 below) in conjunction with the above mathematical properties give rise to the following additional propert-ies: c z = 0 and c R < 0 (3.38) - 62 -mz = c g Z = 0 and £ z = c w Z = 0 (3.41) mR = c g R < 0 and £ R = c w R < 0 (3.42) CZZ > 0 CRR > 0 CZR < 0 ( 3 ' 4 3 ) Conditions (3.37) give the input demand functions in abbreviated form and re-quires that they are positive (monotonicity). Conditions (3.38) give the static (inverse) shadow prices of the fixed inputs (see section 4.1 below). The shadow price of mining structures is of ambiguous sign reflecting the un-predictability of the effect of fixed capital on variable costs. The shadow price of the resource (or grade) c R is expected to be negative, reflecting the rising costs of mining depleted deposits and/or of treating lower grade ore. Given linear homogeneity, conditions (3.39) ensure concavity of c(-) in variable input prices. Normalization of (3.39) and (3.40) yields the Allen elasticities of substitution between variable inputs, where the nor-malization is chosen so that the elasticities are invariant to scaling. An analogous normalization of'(3.41)'and (3.42) yields the inverse elasticities of intensity between, variable and fixed inputs. Finally a simdilar normal-ization of (3.43) gives the inverse elasticities of substitution between fix-ed inputs. (For a detailed discussion of these concepts see section 4.3 of Chapter 4). Consider, now, the foreign trade transformation function e («) in equa-tions. (3.3). Invoking the 'small country1 assumption which ensures that the country's exports of Q and imports of Y do not affect their respective world prices, p and p^  and deducting payments of foreign inputs (g.m.Q) we can rewrite equation (3.3) as Y2 = [pQ - g.m(g, w;; Z, R) Q]/p (3.44) - 63 -Without loss of genera l i ty we may choose units o f Y so that p y = 1. Then, d i v i d i ng both s ides of equation (3.44) by populat ion E we obtain the per cap i ta quant i ty o f Y acquired in exchange f o r the country ' s mineral exports , Q: y 2 = [p - g . m (g, w; Z, R )J q (3.45) where y 2 = Y^ / E, and q = Q /. E. Given our i n i t i a l general assumptions of sect ion 3.1, equation (3.45) assumes balance of payments equ i l i b r ium. The model could be modif ied to include an e x p l i c i t balance of payments cons t ra in t which would allow fo r accumulation of fore ign debt or a c q u i s i t i o n of fore ign assets . Such a cons t ra in t was not introduced as i t would have fu r ther complicated an already complex m o d e l . 1 7 In Appendix B, however, we show a poss ib le way in which th i s cons t ra in t could be introduced. To obtain the to ta l quant ity of Y in per cap i ta terms d iv ide both sides of equation (3.1) by populat ion E, to obta in ; y = y 1 + y 2 (3.46) Subs t i tu t ing (3.26) and (3.45) into (3.46) we ob ta in , y = <f>(k, h). + [p - g. m (g, w; Z, R) ] . q (3.47) S i m i l a r l y equation (3.4) i f d iv ided by E and rearranged may be wr i t ten in per cap i ta terms as: s = y - x - v (3.48) where s = S / E i s per cap i ta non-mining investment; x = X/E i s per cap i ta consumption, and v = V / E i s per cap i ta investment in mining s t ruc tures . F i n a l l y equation (3.5) may be rewri t ten as fo l lows: h = N / E = B / E - IQ I E = £q (3.49) - 64 -where h is the per cap i ta non-mining l abor , 5 i s the labor force as a f r a c -t ion of the populat ion and 1 i s the cost minimizing demand for labor in min-ing per unit of mineral output. Thus,£Q i s the to ta l mining demand fo r labor and zq i s the per cap i ta mining demand f o r labor. Subs t i tu t ing equa-t ion (3.36) in (3.49) we obtain h = % - A(g, w-; Z, R)q (3.50) where £ ( • ) i s the var iab le -cos t -min imiz ing demand for domestic labor by the mining indus t ry . 3.6 The Complete Model and Opt imal i ty Condit ions: An Economic Interpretat ion The complete intertemporal model cons i s t so f the ob jec t i ve funct ion (3.7); the dynamic const ra ints (3.19), (3.23) and (3.25); the technolog ica l const ra ints (3.26), (3.35), and (3.36); the trade cons t ra in t (3.45); the 'adding up' equations (3.46), (3.48), and (3.49); the i n i t i a l resource condit ions (3.18), (3.22), and (3.24); and the terminal condit ions (3.12) -(3.14). Thus, the p lanner ' s problem i s t o : maximize / J u(x)e ~ p t dt (3.7) q,x,v,w subject t o : f = - q - nr (3.19) 1 = v - ( 6 2 + n) z (3.23) ft = s - ( 6 1 + n) k (3.25) s = y - x - v (3.48) y = y 1 + y 2 (3.46) y , = * ( k , h) (3.26) - 65 -h = c - £q (3.49) a =• A(g, w; Z, R) (3.36) y 2 = [p - gm] q (3.45) m = m(g, w; Z, R) (3.35) r(0) = r Q , z(o) = z Q , k(0) = k Q (3.18), (3.22), (3.28) r (T) = r T , z(T) = z T , k(T) = k T ( 3 J 2 ) _ { 3 U ) x t - 0 , q t - 0 , v t - 0 , w t - 0 , s t - 0 (3.51) where (3.51) are non-negat iv i ty const ra ints on the contro l var iab les and the resource stock. By success ive subs t i tu t i on the model may be reduced to: maximize u(x) e ~ p t dt (3.7) subject t o : k = <|>[k, 5- U w , g; Z, R) q] + [p - gm(w, g; Z, R)]q - x - v - ( § 1 + n)k (3.52) z = v - ( s 2 + n)z (3.19) f = - q - nr (3.23) plus the i n i t i a l , te rmina l , and non-negativt ity:constra\ ints. The necessary condi t ions f o r maximizing the ob jec t i ve funct ion (3.7) subject to the dynamic const ra int s (3.52), (3.19), and (3.23), are derived by forming the current value Hamiltonian, H: H = u(x) + pgk + yz + Tjjf* = u(x) + . eC(y-x-v) - (6-,+n) k] + y[v - (6 2 +n) z ] - \|»[q+nr] or in f u l l , H = u(x) + 3 U [ k , K - i(w, g; Z, R) q] + [p - gm(w,-g; Z, R)]q - 66 --x-v-(6 1 + n)k j + Y {v - (6 2 + n)z } +ty {-q-nr} (3.53) where H may be thought of as the net national product in u t i l i t y terms. The costate v a r i ab l e s , y and 3 are thedsocia l demand pr ices o f a unit o f inves t -ment i n mining and oon-mining c a p i t a l r e s p e c t i v e l y . They are to be compared with each other as well as with t h e i r opportunity cost or supply p r i c e , the marginal u t i l i t y o f consumption obtained from the ob jec t i ve func t i on . There-fore 3 and y must be of the same dimension as the ob jec t i ve func t ion . The t h i r d costate var i ab le ty i s the (dynamic) shadow p r i ce or user cost of the resource. It must be noted that the i n i t i a l values 3, y, and ty are not h i s -t o r i c a l l y given but are assigned by the Planning Author i ty . Assuming an i n t e r i o r so lu t ion e x i s t s , the opt ima l i ty condit ions are derived by employing the Maximum P r i n c i p l e o f Pontryagin e t . a l . [1962]. For the maximization o f the soc ia l welfare funct ion (3.7) subject to the dynamic constra ints (3.52), (3.19), and (3.23), i t i s necessary: ( i ) the Hamiltonian (3.53) i s maximized at each point in time with respect to the contro l v a r i ab l e s , x, q , v, arid w: H x = u x " : e = 0 ( 3 - 5 4 ) H y = - g + y =i.O (3.55) H q = 3 [-<}>,/ + (p - gm)] - ty = 0 (3.56) Hw = 6 C " •hV " 9 V ] =0 <3'57) ( i i ) the costate var iab les 3 andy s a t i s f y the fo l lowing d i f f e r e n t i a l equations: 3 = P3 - H k = R3 - 3[<|>k - (6 ] + n)] (3.58) Y = PY - H z = P Y - 3 [- <()hE £ z q - gEm zq] - y[- (6 £ + n)] (3.59) - 67 -i = P.* - H p = pij) - 3[- <f>nE£Rq - gEmRq] - ^[ -n] (3.60) By subs t i tu t i on and rearrangement of terms equation (3.54) - (3.60) become: 3 = u (3.54) ' Y = 3 (3..55)1 * = (p - gm - <f>h£ )e (3.56) ' Vw = 9mw ( 3 - 5 7 ) ' 3 = [P + n + 6 1 - <j>k]3 (3.58) ' y = [p + n + 62 + (<{>h£z + gm z )g ]y (3.59) ' <f>. A R + gmR fr=[p + n + H V Q ] ^ (3.60) ' p - gm - 4>hV Equations (3.54) ' - (3.57) ' give the optimal rules fo r the so lu t ion of the three a l l o c a t i o n problems posed in sect ion 3.2, while the d i f f e r e n t i a l equa-t ion (3.58) ' - (3.60) ' are zero p r o f i t condit ions d i c t a t i n g the evolut ion of the shadow pr ices over time. The remainder of th i s sect ion i s devoted to the economic i n te rp re ta t i on of these opt ima l i t y condit ions in the above o r -der (not ice that given (3 .55 ) ' , condit ions (3.58) ' and (3.59) ' w i l l give i d e n t i c a l optimal paths f o r 3 and y). Equations (3.54) ' and (3.55) ' give the optimal ru les for the a l lo; - -c i i o n of the homogeneous commodity, y , between current consumption, i nves t -ment in mining s t ruc tu re s , and investment in non-mining c a p i t a l . Opt imal i ty condi t ion (3.54) ' requires that the demand pr i ce of non-mining investment (or shadow p r i c e of k ) , 3, be set equal to investment ' s .opportunity cos t , the marginal u t i l i t y o f foregone consumption, u . Thus, i f 3 < u v the p l an -A A hers should a l l o c a t e the marginal un i t o f , y to current consumption, whi le i f 3 > u , they should a l l o c a t e i t to non-mining investment. Opt imal i ty con-A - 68 -d i t i o n (3 .55 ) ' , on the other hand, requires that the demand p r i ce 3 and y of the two type investments (that i s , the shadow pr ices o f the'two -capital stocks) should be equa l ized. Thus, i f y < 3 the planners should a l l o c a t e the marginal unit o f y to b u i l t up non-mining cap i t a l while i f y > 3 more mining s t ructures are de s i r ab le . Combining (3.54) ' and (3.55) ' we obtain the f u l l a l l o c a t i o n r u l e : ¥ = 3 = u x (3.61) which requires equa l i t y of marginal benef i t s der ived from a l t e r n a t i v e uses of y . Given condit ions (3.8) and (3.10) on the u t i l i t y f unc t i on , u > 0 and A £im u = 0 , f o r x < » we have u > 0 and hence y > 0 and 3 > 0, that i s , the shadow pr ices of mining and non-mining cap i t a l stocks are s t r i c t l y p o s i -t i v e . The opt ima l i t y cond i t ion (3.56) ' const i tutes the production ru le f o r mineral output, q, or a l t e r n a t i v e l y the a l l o c a t i o n ru le f o r the resource, r, between the present and the fu ture . Consider f i r s t the economic i n t e r p r e t -at ion of the ind iv idua l terms in equation (3 .56) ' . The world p r i ce of r e -f i ned metal , p, gives the marginal and average revenue of q. The product, gm, of the average quant i ty o f imported input , m, and i t s s e rv i ce p r i ce g, gives the marginal and average cost o f q in terms o f imported inputs . F i n -a l l y ifelae product, q)^£, of the average quant ity of mining l abor , 1, and i t s opportunity cos t , c ^ , gives the marginal and average cost o f q in terms of domestic inputs , i . e . in terms of the non-mining output foregone in order to produce one uni t o f q. Since p i s the marginal revenue and (gm + <j>^ £) the marginal cost of q , the d i f f e rence between the two, p - (gm + <j>^£), i s the net marginal revenue of q, and 3 (= y = u v ) i s the soc ia l va luat ion of th i s - 69 -revenue. Hence, the optimality condition (3.56) states that the optimal rate of extraction must be chosen so that the marginal social benefit from current output, (p - gm - <J>^ JI)B, equals the marginal user cost of the re-source, ty. In terms of the second intertemporal problem posed in section 3.2 this implies the balancing of present benefits against future benefits foregone. Alternatively the production rule (3.56)' may be written as: MR = MC + ty/& (3.62) This is the, monopolist's rule for transforming exhaustible resources into reproducible capital. The optimum rate of extraction is reached where mar-ginal revenue (MR) exceeds marginal variable cost (MC) by just the amount necessary to cover the marginal user cost of the resource. This is analo-gous to the result obtained by Hotelling [1931] except for the term 3. Since in our case the resource revenues are used in building reproducible capital (or in raising the standards of living), the user cost is adjusted by the social demand price of investment, 3 = y . The higher 3 or y, i.e. the more development-oriented the society is, the smaller the 'effective' user cost of the resource, since rapid accumulation of reproducible capital partially compensates for the depletion of the natural capital. Given competitive conditions in the world metal market (or our 'small country' assumption) and constant returns to variable inputs in the produc-tion of q , 1 7 a equation (3.62) may be written as follows: P - AC = ty/$ (3.63) This.states that the country should keep producing as long as i t covers its average variable cost (AC) after making a due allowance for the exhaustibil-- 70 -i t y o f the resource; an average va r i ab le p r o f i t equal to i/>/3 must be earned •j o to cover the e f f e c t i v e user cos t . This i s a s t r i c t l y po s i t i ve amount s ince 3 > 0 and > 0 (given that the resource i s a normal good and the country ' s stock f i n i t e ) . Opt imal i ty cond i t iond(3.57) 1 sets the optimal ru le f o r the a l l o c a t i o n of domestic labor between the mining and non-mining sec tor s . Consider the ind iv idua l terms: 1 i s the f a l l in the mining demand fo r labor and m the w 3 w r i s e in the mining demand fo r imported inputs due to a small increase in the wage r a t e ; <j>^  i s the marginal product of labor in the non-mining sector and g i s the se rv i ce p r i ce of imported mining inputs . Hence, a small increase in the mining wage rate w i l l lead to a gain of i j > ^£ w in terms of increased domestic production of y , and a loss of-gm,, in terms of increased cost of imported inputs and therefore decreased imports of y . It fol lows that an optimal wage po l i c y w i l l set < j )^£ w = gmw > Such a po l i cy w i l l ensure an o p t i -mum a l l o c a t i o n o f domestic labor between i t s two uses. As noted in sec t ion 3.2 th i s a l l o c a t i o n is not an intertemporal problem. As a r e su l t no dynamic shadow pr i ce appears in equation (3 .57 ) ' . Indeed, the ' s t a t i c shadow pr i ce of l a b o r ' , <j>^, can be e l iminated from the dynamic opt ima l i ty condit ions by so lv ing f o r <j>^  in equation (3.51) ' and s u b s t i t u t i n g , ^ into the other o p t i -mal i ty condit ions (3 .56 ) ' , (3.59) ' and (3 .60 ) ' . For instance, the equation f o r (3.56) ' now becomes: * = [P - g(m - Am w /* W )]B • (3.64) The d i f f e r e n t i a l equations (3 .58 ) ' , (3.59) ' and (3.60) ' are zero pro-f i t condi t ions d i c t a t i n g the evo lut ion of the three shadow pr ices over time. Equations (3.58) ' and (3.59) ' determine the rate o f c ap i t a l gains required at each ins tant in order to persuade the community to hold the ava i l ab le - 71 -stocks o f the two types o f physical c a p i t a l ; s i m i l a r l y equation (3.60) ' i n -d icates the rate o f " r e s o u r c e g a i n s " required f o r holding the resource stock remaining in the ground at each ins tant . When so lved, these equations y i e l d the values of the costate var iab les (or shadow pr i ces ) at each point in time, g ( t ) , y[t)a and ty{t), in terms of i n i t i a l values g (0) , y(0) and <JJ(0). According to equation (3 .58 ) ' , g < 0 i f <j>k> p + <S-j + n, i . e . the shad-ow pr i ce o f non-mining cap i t a l would be high and f a l l i n g i f the marginal p roduc t i v i t y of c a p i t a l , 4 ^ , exceeds the sum of the rates of d iscount, de-prec i a t i on and populat ion growth. The reverse i s true i f <j>k < p+ 6-j + n. Since g > 0 i t fol lows that g = 0 i f q>k = p + 6^  + n. This i s a r e su l t s i m i l a r to that obtained from a Ramsey-type model with d iscount ing and cap-i t a l decay. Consider, now, equation (3 .59) ' . As we might expect the eva luat ion of Y over time i s analogous to that o f g except f o r the term ( ^ ^ + gm z)Q. This term, however, i s analogous to ^ in (3.58) ' as i t represents the e f -fec t on the mining var i ab le costs of a marginal change in the stock o f mining s t ruc tures . The sign of y depends on the sign of th i s term which, however, i s ambiguous according to (3.41). I f £^ >^  0 and m z >_ 0 then unambiguously y > 0, while i f £^ < 0 and m z < 0 the sign of y depends on r e l a t i v e mag-nitudes. Since y > 0 i t fol lows that y = 0 i f p + 62 + n = ($ hA z + gmz)Q (3.65) Equation (3.65) impl ies that the shadow p r i ce o f mining s t ructures would not change as long as the decrease in var i ab le mining costs due to a marginal decrease i n the quant i ty o f mining s t ructures exact ly o f f se t s the cost i ncur ? red in terms of i n te re s t foregone and addi t iona l deprec iat ion in per cap i ta s t ruc tures . - 72 -F i n a l l y , the time rate of change o f the shadow p r i c e , \p, of the min -era l resource i s given by (3.60) ' which may be rewrit ten as: <f>.£R + gm R 0 » = P» + n» + [ p h _ K g m (3.66) The f i r s t term o f the RHS of (3.66) ind icates that the marginal user cos t , \i>, grows at the rate of i n t e r e s t . This i s the conventional r e s u l t obtained by Hotel l i n g [1931], Scott [1967] and others f o r the cons tant -cos t - indus t ry or homogeneous-resource case. The second term, nip, takes care of populat ion growth. Assuming a homogeneous resource ( i . e . ignoring the l a s t term), the user cost ip grows at the rate of i n t e r e s t , p, plus the rate of populat ion growth, n. A s i m i l a r r e su l t was obtained by Vajed [1975]. The last , term of (3.66) incorporates the e f f e c t o f resource heterogen-e i t y or increas ing costs due to cumulative product ion. The denominator i s the net marginal revenue which we have already shown to be s t r i c t l y p o s i t i v e . The numerator, on the other hand, ind icates the e f f e c t on va r i ab le cost o f , a marginal change in the resource. From condit ions (3.42) we know that £ R < 0 and mR < 0. Hence, the en t i r e t h i r d term in the RHS of (3.66) i s neg-a t i v e , making the shadow pr i ce i|» grow at a rate lower than p + n by the mar-ginal e f f e c t o f resource deplet ion on ext rac t ion cos t . This r e s u l t i s ana l -ogous to the one obtained by Schultze [1974:64]: "where the use i s sequent-i a l , the p r i ce of mineral r i ghts for the grade cur rent l y extracted w i l l r i s e at a rate less than the i n t e r e s t r a te , r e f l e c t i n g the increas ing cost o f ex-t r ac t i on ." Another i n t e r e s t i n g r e su l t i s the re l a t i on sh ip between the shadow pr ices o f natural cap i t a l and physical cap i t a l in mining. Subs t i tu t ing (3.55).' into (3.56) ' we obta in : - 73 -ty = (p - gm - $ hA) Y (3.67) where (p - gm - jj>^) i s the net marginal revenue from mineral exports. Further, sub s t i t u t i ng (3.67) in to (3.60) ' and rearranging both (3.59) ' and (3.60) ' we have: y = (p + n + 62) y + Q ( + g m Z ) Y ( 3 ' 6 8 ) ty = {p + n)ty + Q(<f»h^ R + gm R ) Y (3.69) The f i r s t term of (3.69) does not include a deprec iat ion rate s ine the min-eral resource does not depreciate as a r e su l t of. the mere passage of time whereas s t ructures do depreciate with time. The second term of (3.69) i s not only completely analogous to (3.68) but is a lso evaluated at the same shadow p r i ce Y- This, s t r i k i n g s i m i l a r i t y between y anci ij> should not be su rp r i s i n g . Both Z and R are cap i ta l goods which enter as f i xed inputs into the mining technology and act as s h i f t parameters on the var iab le ext ract ion cost . The f a c t that R i s non-renewable cap i ta l while Z i s reproducib le i s r e f l e c t e d in the f i r s t term of (3.69) where ty appears instead of y. Of course, ty i s r e l a t e d to y through (3.67). The second term in equations (3.68) and (3.69) simply expresses s h i f t s in var iab le input demand funct ions due to f i x e d input changes. These s h i f t s are evaluated in terms of t h e i r opportunity cost cons i s t ing of investment and consumption foregone or gained s ince y = 3 = u v . The presence of three stock va r i ab l e s , K, z , and r, precludes the use of two dimensional phase-diagrams to p lo t the time paths of the stocks against t h e i r shadow p r i c e s . In f a c t , not much more can be sa id without a f u l l numerical so lu t ion to the model. Numerical s o l u t i on s , however, depend c r i t i c a l l y on the values of : ( i ) the exogenously given parameters, p , n, 6^, 6 2, p, g and S ; ( i i ) the parameters o f the u t i l i t y funct ion in the maximand&j (3 ,7) ; and ( i i i ) the parameters o f the production technolog ies, (3.26), and (3.34). Once a bas ic numerical so lu t ion i s obtained the s e n s i t i v i t y of the numerical r e su l t s to the assumed values of the exogenous parameters may be tested by varying the parameter values around t h e i r bas ic so lu t i on l e v e l s . A s i m i l a r s e n s i t i v i t y ana lys i s may be c a r r i e d out f o r the parameters of the u t i l i t y func t ion . Given a single-consumption-good model and the convention- . al assumption of add i t i ve s e p a r a b i l i t y over time, the intertemporal u t i l i t y funct ion may beerepresented by the in tegra l o f instantaneous discounted u t i l i t i e s , each of i den t i c a l funct iona l form u(x^). Since u(x^.) i s a s ing le argument u t i l i t y f unc t i on , any funct iona l s p e c i f i c a t i o n and any set of para-meters that preserve the regu l a r i t y condi t ions of monotonic ity, and concav-i t in x^ are admiss ib le. Of course, the degree of i n s e n s i t i v i t y o f the num-e r i c a l r e su l t s to va r i a t i ons of the u t i l i t y parameters (and the funct iona l form) must be determined (see Chapter 8 below). The ro le of the t h i r d group of parameters ( technolog ica l ) in numerical so lut ions i s important and complex enough to warrant a separate treatment. 3.7 The Role of Production Technologies Even in h ighly aggregative models, production technologies are m u l t i -f ac to r r e l a t i on sh i p s . To assume a r b i t r a r y values fo r theriir parameters to obtain a bas ic numerical so lu t ion and then attempt s e n s i t i v i t y ana lys i s w i l l not only be an almost meaningless exerc i se , but a l so the computational bur -dsn would be enormous. The conventional p r a c t i c e , fo l lowed i n the per t inent 19 l i t e r a t u r e , i s to spec i fy simple funct iona l forms such as the Leont ie f , - 75 -Cobb-Douglas or the CES funct ions and to estimate t h e i r parameters using h i s t o r i c a l time s e r i e s . These funct iona l forms, however, are known to im-pose a p r i o r i r e s t r i c t i o n s on the e l a s t i c i t i e s o f f a c to r s u b s t i t u t i o n . The Leont ie f technology combines inputs in f i xed propor t ions , thus, a l lowing zero s u b s t i t u t i o n ; the Cobb-Douglas r e s t r i c t s the e l a s t i c i t y o f sub s t i tu t i on to be equal to one. These r e s t r i c t i o n s preclude any ana lys i s of the s e n s i -t i v i t y y of the numerical re su l t s to the assumed technology. To remedy th i s problem Kendrick and Tay lor [1968], [1970] and Chenery and Raduchel [1971] employed the constant e l a s t i c i t y o f sub s t i tu t i on (CES) funct iona l form to carry out s e n s i t i v i t y tests "by varying sectora l e l a s t i -c i t i e s of sub s t i tu t i on while at the same time reca l cu l a t i n g the e f f i c i e n c y parameters.. . to f i t three pieces o f data ( i n i t i a l c a p i t a l s tocks , labor 20 forces and gross production l e v e l s ) " . While th i s ana lys i s may give an i n -d i ca t i on of the s e n s i t i v i t y o f the numerical re su l t s to i n te r sec to ra l chan-ges in subs t i tu t i on p o s s i b i l i t i e s , i t gives no cons iderat ion to the i n t e r -temporal v a r i a t i on of the e l a s t i c i t i e s o f subs t i tu t i on due to r e l a t i v e p r i ce changes over time. Indee'd, there appears to be no way. of t e s t i ng the sensi= t i v i t y o f the numerical resu l t s to the assumed constancy of subs t i tu t i on e l a s t i c i t i e s , short o f spec i fy ing and est imat ing f l e x i b l e production techno-l o g i e s . Before we proceed to such s p e c i f i c a t i o n and es t imat ion, i t i s r e -vea l ing to examine the s p e c i f i c r o l e of the production technologies i n terms of our model-by expressing the opt ima l i ty condit ions in terms of e l a s t i c i t i e s and s t a t i c shadow p r i ce s . We have already seen the dec i s i ve ro le played by the p a r t i a l d e r i v a -t i ves of the production technolog ies , and in p a r t i c u l a r of the var i ab le cost f u n c t i o n , i n the opt ima l i ty condit ions (3.54) ' - (3 .60) ' . By subs t i tu t i ng - 76 -equations (3.57) ' into ( 3 .56 ) ' , (3.58) ' and (3.60) ' we obtain the extraction rule and the zero profit conditions for mining structures and the resource in terms of the exogenous parameters and the partial derivatives of the vari-able cost function alone. As we noted in section 3.5 these derivatives are related to various elasticities and static shadow prices. In a simple two-rimput production function, the technology may be des-cribed by the Allen [1938] partial elasticities of substitution (AES). Usawa [1962] has shown that AES, a ^ , between inputs i and h, may be defined in. terms of a well-behaved cost function by the following convenient formula: a i h = c . c i h / c i c h for i , h = 1 I (3.70) where c is the cost function, c^  and c^ are f irst partial derivatives with respect to input prices (i.e. the demand functions according to Shephard's [1953] lemma) and c ^ are ecoss-partial derivatives. Diewert [1974] ex-tended formula (3.70) to the case of the variable profit function. In section 4.3 below, following the example of Diewert, we apply (3.70) to the variable cost function to derive, in addition to AES, inverse elasticities of substitution between fixed inputs and inverse elasticities of intensity between variable and fixed inputs. Furthermore, in section 4.1 we derive factor shares and static shadow prices for the fixed inputs, [see formulas (4.6) - (4 .8 ) ] . By appropriate substitution and rearrangement of terms we were able to express the entire system of the optimality conditions (3.54) ' -(3.60) ' in terms of elasticities (a) factor shares (s) and static shadow prices, (w), as follows: 3 = Y = - [ 1 ] xu x x (3.71) * = {p - c .[s_(l - ^ )]} y (3.72) ••ll - 77 -I = (p + 6 1 + n)B - [l]b)kB (3.73) * - / J. r x „ \ • r_ m£ £Z ££ mZT /, Y = (p + <52 + n)y + [s — — : Jffl_Y (3.74) ££ * - (p + n)4. + [s lj. w Y (3.75) ££ where: e = - (du /dx) (x/u ) = - xu /u , i s the e l a s t i c i t y of the marginal X X XX X u t i l i t y of consumption; am ~ c- cg W/ cg- c w» i s t n e cross Allen e l a s t i c i t y of substitution between imported inputs and domestic labor; 2 a.. = c.c,„„/c, . i s the own AES for domestic labor; J6J6 WW W "'ic E ^k' 1 S the s t a t i c shadow price of non-mining c a p i t a l ; w z E CZ°* 1 S ^ e s t a t i c shadow price of mining structures; w = cDQ i s the s t a t i c shadow price of the resource stock; r K s m E gm/c = 9£oe^9£ng i s the share of imported inputs in unit v a r i -able cost; °*Z 5 °'cwZ^cw*cZ ^ s t n e i n v e r s e e l a s t i c i t y of intensity (I EI) between domestic labor, £, and mining structures, Z; amZ ~ c , cgZ^ cg' cZ 1 S the IEI between Z and imported dmputs m; E c . c w R / c w . c R i s the IEI between £ and the resource R; a n = c.c D/c .c n i s the IEI between m and R. mR gR g R There are cl e a r l y some advantagesdin being able to express the revenues alloc a t i o n rule in the form of (3.71), the mineral extraction rule in the form of (3.72) and the zero-profit conditions i n the form of (3.73) - (3.75). F i r s t l y , expressing the allocation rule in terms of the e l a s t i c i t y of the marginal uti l ity of consumption, e , enables us to derive the time profile of the consumption path. Recalling that the RHS of (3.71) is equal to, xu - — * * = u y (3.76) e x and differentiating u v with respect to time t, we obtain A u.. = d { u x ] _ d u x dx " V Y x (3V77) x dt dx " dt "xx where k = dx/dt. Solving (3.76) for u and substituting into (3.77) yield, u = - u • (3.78) X X X ' Since 3 = Y = u we have also, 3 = Y = u and X X £ = Y = _ x = _ x (3 79) 1 X Now, substituting into (3.79) the values of-j| and ^-from the zero profit conditions (3.73) and (3.74) respectively and solving for x we obtain the time profile of the consumption path in two equivalent forms: x = 7 |>K - (P. + ^  + n)] (3.80) or x = | « - * Z * q M V . ( p + ^ + n ) ] ( 3 8 1 ) The differential equation (3.80) may be interpreted as requiring that consumption grows along the optimal path at a proportional rate that de-pends upon the elasticity, e , and the excess of the static shadow price of non-mining capital over the sum of the rates of discount, population growth and depreciation. Secondly, writing the extraction rule in the form (3.72) enables us to express the "effective" user cost, \p/y , in terms of the dif-ference between world price, p, and the average variable cost, c, weighted - 79 -by the Allen elasticities of substitution between variable inputs and the cost share of imported inputs. Thirdly, by expressing the zero profit con-ditions in the forms (3.73)-(3.75) we are able to relate the time profiles of the dynamic shadow prices 3, y, and ty of the stock variable k, z, and r to their respective static shadow prices u>. , u , and w . Equation (3.73) K Z IT may be interpreted as requiring that the dynamic shadow price of non-mining capital declines along an optimal path at a proportional rate depending up-on the excess of its static shadow price over the sum of the rates of dis-count, depreciation and population growth. It is, of course, also possible that too much capital already exists, in which case, we should be eating up capital and 3 should be rising; however, this is unlikely to be the case in a developing economy. A similar interpretation may be given to (3.74) and (3.75) except for one important difference. Since mining structures, z, and the resource stock,-r, unlike k, do not enter directly in the production of y, their contribution is evaluated in terms of their effect on the variable input cost of producing Q, which in turn is exchanged for y. The effect of z and r on m and 1 depends (i) on the intensity of use of z and r in relation to m and 1, which is expressed by the inverse elasticities of intensity; (ii) on the Allen elasticities of substitution between m and a; and ( i i i ) on the cost share of these inputs. Thus, in relating y and ty to their respective static shadow prices (uiz and uy) , the latter are weighted by the relevant elasticities and shares [i.e. the terms in the square brackets of equation (3.74) and (3.75)]. Notice also that co^  on the one hand, and u>z and ay on the other, enter with reverse signs since the latter two have been derived from a cost function. - 80 -Consider, now, the implications of the opt imal ity conditions in the form of (3.71) - (3.75). Since a l l terms but the exogenous parameters vary over time as the re l a t i ve input prices (or the marginal rates of subst ir t.iiion) vary, r e s t r i c t i n g e i ther the shares to be constant (Cobb-Douglas) or the e l a s t i c i t i e s of subst i tut ion to be constant (CES), amounts to f i x i ng part of the var ia t ion in the dynamic shadow prices at some arb i t ra ry mom-entary rate. For instance, consider the implications for the user cost of the resource, l j i r a in equation (3.72) of specifying a simple two-input Cobb-Douglas mining technology: since th i s implies unitary a m and constant s m and a over the planning horizon, changes in the value of over time would r e f l e c t changes in absolute but not re l a t i ve input pr ices. A s im i -l a r case can be made for the e l a s t i c i t i e s of in tens i ty although no other study, as fa r as we know, has as yet incorporatedfsuch e l a s t i c i t i e s in an intertemporal context. I t must also be noted that the problem of a p r i o r i r e s t r i c t i on s on technologies i s not pecul iar to numerical so lut ions. In many optimal resource depletion models, the assumption of unitary or con-stant e l a s t i c i t y of subst i tut ion i s invoked to obtain qua l i t a t i ve resu l t s . See, for instance, Solow [1974], Heal and Dasgupta [1974] and S t i g l i t z r [1974]. 2 1 The extent to which the imposition of a p r i o r i r e s t r i c t i on s on the pro-duction technologies biases the (numerical) results i s an empirical question. If the factor shares and subst i tut ion e l a s t i c i t i e s remain f a i r l y constant over time the use of r e s t r i c ted functional forms may be accepted on the . grounds of s imp l i c i t y . I f , on the other hand, wide variat ions of the e las -t i c i t i e s and shares take place over time f l e x i b l e functional forms for the production technologies must be employed. - 81 -The remainder of the present study is concerned with the funct iona l s p e c i f i c a t i o n of the mining and non-mining technologies introduced e a r l i e r in th i s Chapter a n d t h e i r empir ica l est imation using Zambian data. Specia l a t tent ion i s paid to the formulat ion and computation of shares, s t a t i c shadow p r i c e s , and e l a s t i c i t i e s . In p a r t i c u l a r , a f l e x i b l e funct iona l form was chosen which, under appropriate r e s t r i c t i o n s conveniently reduces to a Cobb-Douglas form, wh ich ' i s the most commonly assumed technology in i n t e r -temporal op t im iza t ion . This permits a tes t of the nu l l hypothesis of con-stant and uni tary e l a s t i c i t i e s o f subs t i tu t i on against the a l t e r n a t i v e hypo-thes i s o f p r i ce dependent e l a s t i c i t i e s . In Chapter 8 we return to examine how the estimated technologies might be used in obta in ing a numerical so lu t ion to the model. While the impl i ca t ions of the computed e l a s t i c i t i e s are discussed and the so lu t ion a lgor i thm exp la ined, no f u l l numerical s o lu t i on i s attempted as part o f the present study. To derive i n i t i a l dynamic shadow pr ices f o r a th ree - s tock -model and perform s e n s i t i v i t y ana lys i s on a dozen exogenous parameters, i s a s u f f i c i e n t l y involved s imulat ion pro ject to warrant a separate study. However, the est imation of the technolog ica l parameters and the ana lys i s of subs t i tu t i on p o s s i b i l i t i e s i s a dec i s i ve step towards the d i r e c t i o n of a numerical s o l u t i o n . To quote Chenery and Raduchel [1971:29]: The empir ica l ana lys i s o f subs t i tu t i on p o s s i b i l i t i e s i s therefore c r i t i c a l to the design of planning models and to the i n te rp re ta t i on of t h e i r r e s u l t s . . . Since computa-t i ona l methods f o r so lv ing opt imiz ing models that incorpo-rate production funct ions are now a v a i l a b l e . . . , there is no reason to continue to ignore the p o s s i b i l i t i e s in econ-omy-wide planning. - 82 -FOOTNOTES TO CHAPTER 3 1. Sims and Gotsch [1971:295-296]. l a . Throughout th i s study we use the term economic development as def ined by Adelman [1971:1]. See Introduct ion. 2. For ins tance, Zambia in 1-970 na t i ona l i zed 51 percent of the copper i n -dus t ry ' s assets , reserv ing f o r i t s e l f a dec i s i ve ro le in investment and expansion dec i s i on s , while leaving day-to-day management to Roan Se lec t ion Trust and the Anglo American Corporat ion, now amalgamated into Roan Consol idated Mines and Nchanga Consol idated Copper Mines re spec t i ve l y . (See sect ion 2.4). 3. The "small country" assumption is not t o t a l l y u n r e a l i s t i c fo r most minera l -export ing count r ie s . For ins tance, no s ing le copper exporter has large enough share of the market to have a pronounced long-run e f f e c t on the world p r i ce of copper. As we b r i e f l y discussed in sec-t i on 2.5, above, and expla ined fu r ther in footnote 5.3 to Chapter 2, not even CiPEC has adequate monopol is t ic power to in f luence copper pr ices in the long-run. 4. Subsistence output i s excluded fo r a number of reasons: ( i ) i t cannot be e a s i l y taxed away and red i s t r i bu ted by the Government ( i i ) i t can= not be e a s i l y measured and ( i i i ) i t may be thought of as a lower bound on consumption, which need not be taken e x p l i c i t l y into account i f i t has remained f a i r l y constant over time (in per cap i ta terms). 5. See Appendix B on how th i s assumption might be re laxed. 6. See Appendix B on how an e x p l i c i t balance of payments cons t ra in t might be formulated, to take account of c ap i t a l imports and cap i ta l exports. 6a. See footnote 9a below. 7. For instance as Drysdal l [1972] repor t s , in the case o f Zambia "156,-000 square miles - 54 percent of the country - were prospected in de-t a i l , that i s , by systematic s t r a i g h t - l i n e foot traverses usual ly spaced at quarter mile i n t e r v a l s " as ear ly as the 1930's (see sect ion 2.2 of Chapter 2). 8. This was c e r t a i n l y the case in Zambian copper mining. For ins tance, in Roan Antelope Mine the average grade o f ore mined has f a l l e n from 3.65 percent in 1932 to 1.95 by 1957, and in Mufu l i ra mine from 5.19 percent in 1934 to 2.86 by 1957. (See Table 5.8a below),. 9. The transformation funct ion © (• ) includes the terms of trade and the exchange ra te . 9a. Any balance of payments d i s e q u i l i b r i a are suppressed by the Planners v ia the d i r e c t contro l o f imports o f Y. The d o l l a r value of imported Y i s not permitted to exceed the d o l l a r value of exports minus;: the d o l l a r - 83 -value of imported mining inputs. The domestic value (say in Kwachas) o f imported Y i s obtained by d i v i d i ng i t s d o l l a r value by the ( f ixed) exchange ra te . (While 1 Kwacha = U.S. $1.40, fo r i m p l i c i t y we may assume Kl = $1). 9a. We are aware of the problems involved in the construct ion of such-a soc i a l welfare funct ion (S.W.F.) from ind iv idua l preferences, as demon-s t ra ted by Arrow [1951] and others . We circumvent these problems by assuming that the planners proposed the S.W.F. and the voters approved the proposa l . In t h i s sense S.W.F. may be regarded as r e f l e c t i n g the preferences of the soc ie ty in general (see Burmeister and Dobell [1970]: 335] f o r a s i m i l a r argument). While th i s may not be p e r f e c t l y s a t i s -f a c to ry , our concern i s with the attainment of some predetermined ob-j e c t i v e i t s e l f . It i s not pretended that the postulated S.W.F. repre-sents the ob jec t i ve funct ion of the planners in Zambia or in other count r ie s . 9b;. By;y- ' p re ferences ' we mean the preferences o f the soc ie ty (see footnote 9a) as conceived by the planners. 10. Heal [1973:254]. 10a. Ibid: 254-255. 11. See Kendrick and Tay lor [1971:461], 11a. Ib id. 12. The " c u t - o f f " grade i s used, here, in a geologica l sense not in an econ-omic sense. More appropr ia te ly i t may be c a l l e d the "resource c u t - o f f grade" as opposed to the "reserves c u t - o f f grade." See sect ion 5.5. 13. We assume that any problems of compensation have been reso lved. 14. Toobe exact , the normal izat ion is chosen so that the e l a s t i c i t y a.^ i s i nva r i an t to sca le changes in units and so that a.^ = o^-. Analogous q u a l i f i c a t i o n s apply to a l l normal izat ions re fe r red to below. 14a.It i s assumed that r e f i n i n g takes place in the same loca t ion as mining (see Chapter 2). 15. To quote Banks [1974:84] fo r the case o f Zambia " . . . w h i l e i t was p o s s i 1 . tble f o r the Zambian government to take over the mines, they were not able to organize t h e i r operat ion. As i t was,.they took control o f 51 per-cent of the assets o f the mines.... Arrangements were a l so f i n a l i z e d fo r a management and consultancy contract between the mining companies and the Zambian government. 15a.Zambia, f o r ins tance, adopted an e x p l i c i t " p o l i c y on mineswages." The po l i c y can be b r i e f l y s ta ted: "the government supports worker's demands f o r wage increases only in so fa r as the rate of these increases i s match-ed by a comparable r i s e in the leve l s of output per man... It fears that - 84 -wage increases will lead to capital substitution for labor, with debil-itating effects on the employment targets posited in the National Development Plan" (Bates [1971:36-37]). 16. See for instance Shephard [1970] and Diewert [1974]. 16a. Constant returns to variable inputs are set to prevail when a propor-tionate change in all variable inputs, holding all fixed inputs con-stant, leads to an increase in output by the same proportion (that is, the underlying production function is linearly homogeneous in variable input quantities). Constant returns to variable inputs imply increas-ing returns to all inputs (that is, increasing returns to scale) which may not be an unreasonable assumption for the case of mining. Alter-natively constant returns to variable inputs may be justified on the grounds that the mining industry operates in a range in which the fixed inputs are not fully utilized. (For more details on this point see Appendix B). The possibility of introducing decreasing returns to vari-able inputs (constant returns to scale) and the resulting implications for our model are also discussed in Appendix B. 17. Even as i t is the model is quite complicated as there are three states and four control variables. 17a. Constant returns to variable inputs coupled with no control over the price of q may result in 'insufficient' convexity, in q and consequent indeterminacy of the optimal path of q. This cannot be known in advance of a complete (numerical solution) because of the complexity of the model. Notice for instance that the Hamiltonian is not linear in q as i t appears from the fact that its gradient with respect to q (equation (3.56)') does not involve q. However, <|y does involve q: <fy = <|y [k, 5-$(w,g;Z,R)q]. In Appendix B it is shown how diminishing returns to vari-able inputs might be introduced in the event of 'insufficient' convexity in q. 18. A similar result was obtained by Scott [1967]. 19. For instance, Kendrick and Taylor [1971] and Vajed [1975] use Cobb-Douglas functional forms white Kendrick and Taylor [1968], [1970] and Chenery and Raduchel [1971] employ CES production functions. Earlier studies by Chenery and Kretschmer [1956] Bruno [1966] and Eckhaus and Parikh [1968] use Leontief-type technologies in linear programming models. 20. Kendrick and Taylor [1968:231]. The term 'efficiency parameter' in the quote refers to the multiplicative constant of the CES production function. 20a. Notice that while we transformed the static shadow prices (w) into per capita terms such transformation was not made for the elasticities (a), since the elasticities, unlike the shadow prices, are invariant to scaling. - 85 -It must be noted, however, that while these authors a lso deal with the optimal u t i l i z a t i o n of exhaust ib le resources, t h e i r problem i s cast in a d i f f e r e n t context. CHAPTER 4 MINING AND NON-MINING PRODUCTION TECHNOLOGIES: FUNCTIONAL SPECIFICATION The empir ica l app l i c a t i on o f the optimal contro l model, developed in Chapter 3 to the Zambian case reviewed in Chapter 2 requ i res : ( i ) f unc t i on -al s p e c i f i c a t i o n ; ^ the mining and non-mining technolog ies, c ( « ) and <(>(•); ( i i ) funct iona l s p e c i f i c a t i o n o f the intertemporal u t i l i t y funct ion u ( « ) ; and ( i i i ) transformation o f a l l continuous time constra ints in to d i s c re te time as data are ava i l ab le only in d i sc re te . fo rm. The d iscuss ion of the l a t t e r two requirements i s postponed un t i l Chapter 8 in which the p o s s i b i l -i t y of a numerical so lu t i on i s explored. The purpose o f the present Chapter i s the funct iona l s p e c i f i c a t i o n of the production technologies c ( - ) and <{>(•)• In sect ion 4.1 a t rans log v a r i -able cost funct ion is- s p e c i f i e d f o r the mining sector and in sect ion 4.2 a t rans log production funct ion f o r the res t o f the economy. F i n a l l y , i n sect ion 4.3 we introduce some fur ther theore t i ca l concepts, such as the pr i ce and subs t i tu t i on e l a s t i c i t i e s , through which the technologies to be estimated can be convenient ly descr ibed. 4.1 Mining Sector: A Translog Var iab le Cost Function Our search o f the l i t e r a t u r e uncovered no s a t i s f a c t o r y funct iona l form f o r a mining cost funct ion with an underlying theore t i ca l framework. There i s a growing l i t e r a t u r e on cost funct ions in general but t h e i r app l i c a t i on to mining requires a number o f modi f icat ions to account fo r features p e c u l -i a r to mining. F i r s t , we must d i s t i ngu i sh between two d i f f e r e n t types of reproducib le phys ica l assets : machinery and equipment on the one hand, and bu i ld ing and - 87 -engineeering s t ructures on the other. Machinery and equipment can, in p r i n -c i p l e , be rented and i f purchased can be reso ld without undue loss and, in genera l , can be var ied f r e e l y in the short -run (one y e a r ) . In contract bu i l d ing and engineering s t ructures such as mine sha f t s , mine roads ,mi l l s and smelters, have low or zero resa le value and cannot be var ied f r e e l y in the shor t - run ; they may be l e f t to depreciate but in general cannot be wi th -drawn without substant ia l loss as, in f a c t , they cons t i tu te sunk costs . As a r e s u l t , unl ike the investment in machinery and the h i r i n g of l a -bor, investment in s t ructures and new mine development are not part of the short -run cost -minimiz ing ca l cu l a t i on s of the mining f i rm. The opening of new mines and ithe construct ion of new m i l l s and minetowns involves the commitment of lumpy investment expenditure which can only be based on long-run expectat ions regarding the general economic and p o l i t i c a l c l imate. To emphasize the above d i s t i n c t i o n between the two d i f f e r e n t forms of assets we w i l l be r e f e r r i n g to machinery and equipment as var i ab le cap i t a l and to s t ructures as f i xed c a p i t a l . In developing a cost funct ion f o r min-ing we attempt to expla in only the var i ab le cost of production defined as the sum of the wage b i l l f o r labor and the rental b i l l f o r var iab le cap-i t a l . Var ia t ions in operat ing cos t , however, are not expla ined by v a r i -at ions in output and the pr ices of var iab le inputs alone as the f i xed f a c t -o r s , act ing as s h i f t parameters, a f f e c t the var iab le input requirements per unit o f output. Structures are f i xed not i n the sense of 'a constant num-ber 1 but in that t h e i r va r i a t i on i s not a choice var iab le of the f i rm in the short r u n . 1 A. second feature of mining not shared by other indus t r ie s i s that i t r e l i e s heav i ly on inputs , not made to user ' s s p e c i f i c a t i o n s , but provided - 88 -2 by nature in fixed quantity and location and in heterogeneous quality. In a given mineralized area the amount of copper metal, Aq, found in mineral concentrations higher than that of the common rock is not only limited,but, more importantly, is contained in ore deposits in which there is a gradua-tion from relatively rich (high grade) to relat ively lean (low grade) mat-e r i a l . Assuming the contents of the ore deposit "can be separated into chunks of separate grade, the best grades wil l be taken before the poorer; 4 the best areas before the poorer; and the best mines before the poorer." Hence, the effect of each year's production is to make the remaining mater-ial more costly to extract and treat, through worsening of the working con-ditions in the mine and recourse to lower grades. As this is a cumulative process, and a direct result of cumulative production i t can be best repre-t. sented by cumulative copper output or alternatively by the unexploited t 1 stock of the resource, R. = A - JQ., where A is the i n i t i a l copper endow-t o ? I o r r ment of the afcea concerned. Introduction of R^  as a fixed factor in the short-run cost function summarizes the presumption that as the resource stock is run down its quality deteriorates and its accessbi i l i ty declines, 5 raising the input requirements (or cost) per unit of output. The effect of fa l l ing grades on production costs may be represented 5a more directly by the use of the average grade of cumulative production G t . HQwever, there is a certain advantage in using some form of cumulative out-put, as i t may also serve to capture the 'learning by doing' phenomenon which is l ike ly to be rather signif icant in the case of mining operations in developing countries where highly sk i l led expatriate labor is used side by side with unskilled domestic labor.^ Experience gained in the production process increases the stock of specialized sk i l l s thus lowering the factor - 89 -requirements per un i t o f output in succeeding per iods. This stock of ex-perience i s a ' s c a r ce ' resource which can only be created by the act of pro-duction i t s e l f , and hence i s f i x e d at any given time by the amount of pro -duction undertaken in the past i . e . the cumulative output to that t i m e . 7 'Learning by doing ' and f a l l i n g grades work in opposite d i r e c t i o n s , but the former i s not expected ' to o f f s e t the l a t t e r except f o r the e a r l i e r years when labor i s s t i l l inexperienced and high grade ore i s s t i l l a v a i l a b l e . In order to incorporate f i xed factors in exp la in ing short -run cos t , we employ some resu l t s of the dua l i t y theory to develop a . ' v a r i a b l e ' cost q funct ion which i s analogous to the var iab le p r o f i t func t ion . The mining industry i s assumed to have the fo l lowing short -run pro-duction func t i on , Q = F(V;X) (4.1) • r e l a t i n g output Q to a vector o f va r i ab le inputs V = (V-|,...,Vj) >_ 0 and a vector of f i xed inputs X = ( X l s . . . , X j ) >_-0 . The technology F (« ) i s assumed to be a non-negative, quasi -concave, non-decreasing continuous funct ion de-f ined over the non-negative orthant; Input and output l eve l s are rates o f 9a serv ice flow per un i t of time. The d i f fe rence between var iab le and f i xed inputs i s that the former can be var ied f r e e l y in the short run while the l a t t e r cannot. By fu r ther assuming cost minimizing behav i ou r 1 0 on the part of the min-ming i n d u s t r y 1 1 and competit ion in the input markets, we obtain the fo l l ow-ing var iab le cost funct ion corresponding to the production funct ion (4.1) C = C(P;X,Q) = min{P T- V: F(V;X) > Q} (4.2) V 1 p where C stands f o r t o t a l mining cost , P = (P^ , . . . ,P j ) i s a vector o f s e r -v ice pr ices fo r the I va r i ab le factors of product ion, and T ind icates vector - 90 -transposition. Under the assumptions on F(«), the cost function C(«) is a non-negative continuous function defined over the positive price orthant, non-decreasing in input prices and output levels, linearly homogeneous and 13 concave in variable input prices, and convex in fixed input quantities (regularity conditions on variable cost functions). Proofs of concavity in P and convexity in X fol low: 1 3 9 (i) Proof of concavity in P Note that -C is a variable profit function, i .e., -C(P,X,Q) = maxv {PT (-V) : (Q,-V,-X)|T} Thus from the properties oi&fthe variable profit function we can deduce that -C is convex in prices?!?, and is concave in -X,Q i f T is a convex cone^.is concave in prices. A direct proof of concavity in prices property is given below. C(P,X,Q) E minv {PTV :F(V,X) > Q} let C(P1,X,Q) = P 1 TV 1:F(V 1,X) > Q and C(P2,X,Q) = P 2V:F(V 2,X) > Q Then CUP 1 + (1 -x)P2,X,Q) = (XP1 + ( l -x)P 2 )V = XP 1 TVA + (1-X)P 2 V > XP1 TV ] + (l-x)P 2 TV 2 = AC(P1,X,Q) + (1-X) C(P2,X,Q) ; (ii) Proof of convexity in X Let C(P,X],Q) = P^ 1 \ W >_tj F i s quasi-concave and C(P,X2,Q) = PTV2 , F(V 2,X 2) > Q Then C(P,XX] + (l-x)X2,Q) = PTVX , F(VX,xX ] + (l-x)X 2) > Q < PT(XV] + (1-X)V2) since F(xV1+(l-x)V2,XX1 9 + (1-X)X ) >_ Q by quasi-concavity of F 1 2 i.e. XV +(l-x)V is feasible but not neces-sarily optimal for the cost min. problem: - 91 -min v (P TV : F f V . X X ^ O - A j X 2 ) > Q} = ,XPV + ( 1 - A ) P T V 2 = AC(P,X \ Q ) + (1 -x ) C (P , X 2 , Q ) i . e . C(P ,X,Q) i s convex in X. The conventional "p r ima l " approach to obta in ing input demand equations f o r est imat ion purposes i s t o , f i r s t spec i fy a funct iona l form for the pro-duction funct ion ( 4 J ) ; second, solve the cost minimization problem (4.2) by d i f f e r e n t i a t i n g the corresponding Langrangian funct ion with respect to var iab le input q u a n t i t i e s ; and t h i r d , solve the re su l t i n g f i r s t order con-d i t i ons f o r the optimal demands of var i ab le inputs as funct ions of the ex-ogenousTy given output l e v e l , va r i ab le input p r i ces and f i x e d input quan-t i t i e s . Unfortunate ly, th i s becomes a cumbersome i f not impossible approach i f we want to use a f l e x i b l e funct iona l form fo r F (« ) with the minimum of a p r i o r i r e s t r i c t i o n s on the estimated technology. 14 Dual i ty theory, however, t e l l s us that i f f irms minimize costs then the cost f unc t i on , C(«)» i t s e l f contains enough information to completely descr ibe the technology. The "dua l " approach implies that we need not spec-i f y a funct iona l form f o r the production funct ion to solve the cost minimiz-at ion problem. We can, ins tead, spec i fy a funct iona l form for the cost fun -c t ion C(«) . Provided that the chosen form s a t i s f i e s the regu l a r i t y cond i -t ions oh cost funct ions , given above, C(«) incorporates a l l technolog ica l and economic information contained in the production funct ion and the cost minimizing behaviour. Out of the c lass o f f l e x i b l e funct iona l forms ( i . e . the forms that do not impose a p r i o r i r e s t r i c t i o n s on technology) the transcendental l o g a r i t h -15 mic (or trans log) funct iona l form was chosen. Its app l i ca t i on to the v a r i -- 92 -able cost funct ion C (« ) was made fo l lowing Diewert 's [1974] genera l i za t ion of the var i ab le p r o f i t f u n c t i o n . 1 6 The Translog Var iab le Cost Function may be wr i t ten as: I , I I £nC(P ;X,,Q ) = a n + £ a . £nP . + I ^ - h ^ h J , J J + I a . £nX , + j I I a , . £ n X £nX. j=l J J \ j=l k=l J K J k I J + & ^ 6 i J £ n P i £ n X J (4.3) + e AnQ £nQ£nQ * £nQ l e .£nP. H c HH ^ = - j q 1 1 + -^nQ I C Q i £ n X , j=l q j J where a ' s , a ' s , 6 's, e ' s and c ' s are parameters to be est imated. Equation (4.1) i s homogeneous of degree 1 in var i ab le input p r i c e s , P.,-» i f the fo l low-ing r e s t r i c t i o n s ho ld: I i | 1 6 i j = 0 f o r j = 1 , . . . , J I ( i i i ) I a i n = 0 f o r i = l.-.-.I fi=l I ( i v ) I Q = 0 , and i=l V (v) a.. = a, . , 5 . . = $ . . , ' and e . = e. • lh hi i j y j i qi iq - 93 -Equation (4.3) i s l i n e a r l y homogeneous in f i xed input quant i t ie s X. J and output Q i f : J (vi) I «• + e = i=l J q 1 for q = 1 (vii) J I 6. . + 0 . j=l 1J V = 0 fo r i = 1,. . . , J. , q = 1 ( v i i i ) J k=Vjk + ?qj = 0 fo r j = 1,. . . , J ; q = 1 (ix) J y x, . + © j i ^ q j qq = 0 fo r q = 1 , and (x) 5. . = 6 . . , C 1J J1 5 qi = 9 i q ' V = a, . and c . qj Res t r i c t i ons (v i ) - ( ix) on C (« ) imply l i n e a r homogeneity in a l l inputs o f the underlying production funct ion ( i . e . F (« ) exh ib i t s constant returns to s c a l e ) . Equation (4.3) may be made l i n e a r l y homogeneous in output Q alone by rep lac ing r e s t r i c t i o n s ( v i ) - ( ix) by the fo l lowing : (v i )a = 1 f o r q = 1 ( v i i ) a V ==60 f o r i = 1,...,I; for q = 1 (vi i i ) a = 0 fo r j = 1 J : fo r q = 1 ( ix )a 6 qq = 0 fo r q = 1 (x)a »qi = 9 i q ? q j = ^jq Res t r i c t i ons (v i )a - (x)a on C (« ) imply l i n e a r homogeneity in var iab le i n -puts o f the underly ing production funct ion ( i . e . F ( « ) exh ib i t s constant r e -1 fia turns to va r i ab le inputs when the f i xed inputs are held constant or i n -creas ing returns to sca le in a l l i nput s ) . Even i f a l l r e s t r i c t i o n s ( i ) - (x) or ( i ) - (v) and ( v i ) a - (x)a are imposed there s t i l l remain enough f ree parameters to provide a second order - 94 -approximation in output, var iab le input pr ices and f i xed input quant i t ie s to an a r b i t r a r y twice cont inua l l y d i f f e r e n t i a t e cost funct ion that s a t i s -f i e s the appropr iate homogeneity p roper t ie s . While the above r e s t r i c t i o n s can be e a s i l y imposed a p r i o r i on the va r i ab le cost funct ion (4.3) to assure attainment of the des.ired homogen-e i t y p roper t i e s , i t i s qu i te d i f f i c u l t to obtain s u f f i c i e n t condit ions on the parameters o f a t rans log funct iona l form to ensure that a l l r e gu l a r i t y condit ions on cost funct ions ( i . e . condit ions on C (« ) in equation (4,2))are g l oba l l y s a t i s f i e d . The only exception is the spec ia l case obtained i f , in add i t ion to symmetry and homogeneity, the r e s t r i c t i o n s a ^ = 0, i ,h = 1,...,I i s imposed, in which case (4.3) co l lapses to a Cobb-Douglas form in var i ab le input p r i c e s . I f, however, at l eas t one a ^ f . 0 both monotonicity and con-vexi ty may b e - v i o l a t e d , because of the quadrat ic nature of the t rans log fun-ct ion&Teform.(S imi lar ly concavity i n f i x e d inputs^may be v i o l a ted i f at l eas t one a-, f 0 ) . This may not be a ser ious problem in empir ica l app l i ca t ions as Woodland [1976] noted: " . . . while global r e g u l a r i t y may be des i rab le i t i s not es sent ia l s ince i t usual ly s u f f i ce s that the funct ion be regular in the subset of the sample space that i s emp i r i ca l l y r e - l evan t . 1 , 1 7 Thus, the estimated cost funct ion can be checked to determine whether the regu l a r i t y condit ions are s a t i s f i e d at each data po int . Equation (4.3) may be considered e i t h e r as the form o f the true cost funct ion or a second order approximation of the true funct ion of some un-known form, with the approximation e r ro r being taken care of by the s toch-a s t i c s p e c i f i c a t i o n . The present study assumes that equation (4.3) i s the true cost funct ion and proceeds to estimate i t s unknown parameters assuming an add i t i ve e r ro r term, explained in sect ion 6.1 below. D i rect est imation - 95 -of £ ( • ) » however, may invo lve col l i n e a r i t y problems g iv ing r i s e to s t a t i s -t i c a l l y i n s i g n i f i c a n t parameter est imates. It i s preferab le to use, i n -s tead, the f a c t o r demand equations to estimate the parameters o f C ( « ) -In der i v ing the var iab le input demand funct ions from equation (4.3) we employ Shephard's [1953] lemma which states that i f C(•) i s d i f f e r e n t i -able in input pr ices at the point P* the cost-minimizing quant i ty o f i n -* put Vij needed to produce output Q, given input pr ices P' and f i xed quant-i t i e s X, co inc ides with the p a r t i a l de r i v a t i ve of C with respect to the serv ice p r i c e of input i , P., at point V i = C.(P*;X,Q) = 3C(P*;X,Q)8P. i = 1 1 (4.4) or in the case of constant returns to var i ab le inputs , V i = Q • ac( 'P*;X)/3P i (4.5) where c ( « ) i s the unit va r i ab le cost func t ion . It turns out that the cost minimizing demand funct ions in the, case of t rans log are not l i n e a r in- the un-known parameters. It i s then more convenientttooestimate the f ac to r share equations which are l i n e a r in the unknown parameters. The share of f a c to r i in the to ta l cost o f production i s def ined as PiV.(P;X,Q) S.(P;X,Q) = (4.6) I P nV h (P;X,Q) h=l n n The share equations are e a s i l y derived by l oga r i thmica l l y d i f f e r e n t i a t i n g equation (4.3) S^PjX.Q) = 3AnC ( P ;X ,Q)/3AnP. (4.7) - 96 -In empir ica l app l i ca t ions the share equations (4.7) and the cost func-t ion (4.3) must be estimated as a simultaneous system fo r at l eas t two rea -sons: ( i ) to regain degrees o f freedom l o s t by the f a c t that shares sum to unity and ( i i ) to regain the a b i l i t y to reconstruct the cost f unc t i on , which is l o s t in the process o f der iv ing the share equations v ia d i f f e r e n t i a t i o n of (4.3). F i n a l l y , i f the f i xed. inputs are ac tua l l y var iab le ( l o n g - r u n ) , 1 7 3 and the va r i ab le cost funct ion i s d i f f e r e n t i a t e with respect to f i x e d input quant i t i e s at point X , then the p a r t i a l der iva t ives of C (« ) with respect to X. co inc ide with the "shadow p r i c e " or "imputed va lue " , w., o f f i xed i n -J J put X-, i . e . w* = 3C(P*;X*,Q)/3X. f o r j = l , . . . , J (4.8) where X i s a so lu t ion to the fo l lowing cost minimization problem: min{w*TX: C(P*;X,Q) > C(P*;X*,Q): X > 0 (4.9j_ X * that i s X was chosen so as to minimize the cost of producing the given am-, ount o f output with the given amount of var i ab le cost . In the ensuing analy-s i s we w i l l assume that producers do not optimize with respect to f i x e d i n -puts, and hence the system of equations (4.8) w i l l not be used in the econ-ometric est imation of the parameters o f C ( « ) -For the purpose of est imat ing a var iab le cost funct ion fo r the Zamb.ian copper industry we apply equation (4.3) to the case of one output, two v a r i -able and two f i xed inputs . The output i s the amount of r e f i n e d copper, Q .^, produced in year t . The var i ab le inputs are ( i ) domestic labor , L, and ( i i ) imported aggregate input M, which i s a D i v i s i a quantity index obtains ed by aggregating imported machinery equipment and expatr ia te technica l per-- 97 -sonnel . The f i xed inputs are: ( i ) the phys ica l cap i ta l embodied in bu i ld ing and engineering s t ructures (such as mine sha f t s , m i l l s , concentrators and minetowns) denoted by Z^; and ( i i ) the natural c a p i t a l , or copper resource stock R^  remaining i n the ground in year t . Calendar time T i s employed to 1 o represent Hicks-neutra l exogenous technica l progress. Omitting the time subscr ipts fo r c l a r i t y , we rewrite equation (4.3) in terms of the above i n -puts and output as fo l lows: £ n C ( P M , P L ; Z , R , Q,x) = a Q + a M AnP M + a L£nP,_ + \ a M M ( ( £ n P M ) 2 + 1 2 a M L £ n f V n P L + 2 a L|_^ n P l_) + a Z M A n Z j i n P M + a z ^£nZ£nP^ + a R|v j £ n R £ n P M + a R ^£nR£nP^ + 1 2 a z £ n Z + a R £ n R + ^ a z z ( £ n Z ) + a Z R £ n Z £ n R + (4.10) ^ a R R ( £ n R ) 2 + a Q £nQ + \ a Q Q ( £ n Q ) 2 + a Q M £ n Q £ n P M + a g L £ n Q £ n P L + ag Z £nQ£nZ + ag R £nQ£nR + a £nx T Noting that i , h = M, L and j , k - Z, R and denoting q by Q we may t rans la te r e s t r i c t i o n s ( i ) to (x)a i n to : (1) a., + a, = M L 1 ( i i ) ' aRM + a RL = 0 , a Z M + a Z L = 0 ( i i i ) ' aMM + aML = 0 , a L L + a L M = 0 ( i v ) ' aQM + a QL = 0 (v ) ' Symmetry (aML = a LM e - t - c - ) ( v i ) ' a z + a R + a Q = 1 ( v i i ) ' aZM + aRM + aQM = 0 ' a ZL + a RL + a QL = 0 ( v i i i ) a ZZ + aZR + a Z Q = 0 , a R Z + a R R + a R Q = 0 ( ix) aQZ + aQR + aQQ " (x) Symmetry ( a Z M = a( a Q = 1 aQM = 0 a QL = 0 (v11))j aQZ = 0 aQR = 0 ( i x j ; i aQQ = 0 Symmetry. ( a Q M = a, 98 -MZ ' aZR 3 RZ e ' t ' c - ) MQ e . t . c . ) In what fol lows we impose homogeneity of C (« ) in var i ab le input pr ices i i through r e s t r i c t i o n s ( i ) - ("v) and homogeneity of C (« ) in output alone i i ( i . e . constant returns to var iab le inputs) through '(-V:i)\ ac = !,(x).a. S imult -aneous homogeneity in f i x e d inputs and output i s not imposed. Then (4.10) becomes: £nC = a Q + £nQ + £ n P L + a^inP^ - a^inV^ - l a M L ( £ n P M ) 2 + a M L A n V n P L " a M L U n P L ) 2 + a Z M * n Z £ n P M - a Z M «,nZ i lnP L + a R M £ n R £ n P - a R M £ n R £ n P L + a z £ n Z (4.11) 1 2 1 2 + a R £ n R + 2 a z z ( £ n Z ) + a z R £ n Z £ n R + j - a R R ( £ n R ) + a £nx Equation (4.11) can be rearranged into the fo l lowing t rans log unit va r i ab le cost func t ion : • P P P c M l M 2 M £n((:pO = a Q + a M £ n ( p - ) - j a M L ( £ n p-) + a Z M £ n Z £ n ( p - ) P + a R M £nR£n(p^- ) + a z £ n Z = a R £ n R + ^ a z z ( £ n Z ) 2 ' (4.12) 1 2 + a Z R £ n Z £ n R + ^ a R R U n R ) + a £nx 18a where c = C/Q i s the unit or average cost o f production The va r i ab le f a c to r share equations are ea s i l y obtained by l o g a r i t h -mica l l y d i f f e r e n t i a t i n g equation (4.12) with respect to the var i ab le input 99 -pr ices P M • and P L : p SM = IS. = AM " A M L £ n ( p f ) + A Z M * N Z + a R M £ n R p S L = UPTL = 1 - AM + a M L £ n ( p f ) " a Z M * n Z " A R M £ N R <4-14> where i s the share of imported aggregate input (machinery and expat-r i a t e labor) in the uni t va r i ab le cos t , and S L i s the corresponding share o f domestic l abor . Note that shares must add up to one, S M + S L = 1 (4.15.) The var iab le f ac to r demand equat ions, obtained using Shephard's lemma are re l a ted to the respect ive share equations in the fo l lowing manner: m* = 3c /9P M = (c /P M ) • S M (4.16) Z* - 9c /9P L = ( c /P L ) • S L (4.17) * * where m and i are the cost minimizing quant i t i e s of var iab le of inputs M and L r e s p e c t i v e l y , per unit of output given P M , P|_, Z and R. For completeness, we derive the ( inverse) shadow pr ices w z and w R of the f i xed inputs Z and R by p a r t i a l l y d i f f e r e n t i a t i n g the var i ab le cost funct ion (4.12) with respect to Z and R: p W Z = I ^Z + a Z M £ n ( P ^ + a Z Z £ n Z + a Z R £ n R ] ( 4 J 8 ) ^ P * c M w R = R [ l - a z + a R MAn (p - ) + aR Rs,nR + a Z R £ n Z ] (4.19) The rate of Hicks-neutra l exogenous techn ica l change in given by 9C/3T = (c/x)a (4.20) The above shadow pr ices should be in terpreted with caut ion , as margin-al adjustments of the f i x e d factors may not be poss ib le even in the long run. Structures in p r i n c i p l e cannot be run down except through deprec ia t i on , and - 100 -the resource stock cannot be augmented, assuming i t s f u l l s i ze i s known with c e r t a i n t y . Furthermore, they are s t a t i c shadow pr ices in the sense that they do not take account of the f i x e d f a c t o r ' s user cos t , which i s an intertemporal imputed p r i c e . F i n a l l y they are inverse , in the sense that they are der ived from a cost funct ion rather than a production funct ion or var iab le p r o f i t f unc t i on , and hence t h e i r s ign is reversed. A po s i t i ve i n -verse shadow pr i ce fo r the f i xed f ac to r X t e l l s us how much the producer i s w i l l i n g to pay to get r i d o f the marginal un i t o f the f ac to r ( i f he cou ld ) , as i t s presence ra i ses the var i ab le cost of product ion. A negative inverse shadow p r i c e , on the other hand, t e l l s us how much the producer i s w i l l i n g to pay t o ' a cqu i re one more unit of the f i xed input,..since th i s w i l l lower his va r i ab le costs by that amount. Having s p e c i f i e d a funct iona l form fo r the var i ab le cost f unc t i on , econometric est imation of i t s unknown parameters requires an appropriate s tochas t i c framework and data. The data requirements, sources and method-ology are discussed in Chapter 5, whi le the s tochas t i c s p e c i f i c a t i o n i s postponed un t i l Chapter 6. The empir ica l re su l t s are reported and i n t e r -preted in Chapter 7. We now turn to the s p e c i f i c a t i o n of a functionaTzform fo r the non-mining production func t i on . 4.2 Non-mining Translog Production Function In spec i f y ing the production technology $(•) fo r the aggregate non-mining sector of the economy, we make use of the transcendental logar i thmic or t rans log production function, f i r s t suggested by Cr i s tensen, Jorgenson and Lau [1970]. Unl ike s impler forms such as the Leont ie f , the Cobb-Douglas and the Constant E l a s t i c i t y of Subs t i tu t ion funct ions which r e s t r i c t the A l l en p a r t i a l e l a s t i c i t y of subs t i tu t i on to be zero, one, or constant r e -- 101 -s p e c t i v e l y , the t rans log production funct ion imposes no such a p r i o r i r e -18b s t r i c t i o n s . Indeed, the trans log funct iona l form i s a genera l i za t ion of the Cobb-Douglas (1928) production func t ion . Assuming exogenous Hicks-neutra l techn ica l progress, the t rans log pro-duction funct ion r e l a t i n g phys ica l output to input serv ices may be wr i t ten as: I , I J £n Y = £ n a n + J b.£nX. + y I I b. .£nX.£nX. + b,£nx (4.21) u i=l 1 1 - L i=l j=l 1 J 1 J T where Y i s the output l e v e l , aQ i s the s tate of technica l knowledge, X. and X. are serv ices of inputs , T i s calendar time represent ing Hicks-neu-t r a l exogenous techn ica l progress, and b., b.. are techno log i ca l l y deter -mined parameters. The t rans log formulat ion (4.21) requires that a l l input leve l s be s t r i c t l y p o s i t i v e , otherwise output becomes i l l - d e f i n e d . Equation (4.21) is l i n e a r l y homogeneous in input leve l s exh ib i t i n g constant returns to sca le i f the fo l lowing r e s t r i c t i o n s ho ld , IS - 1> I.b.- = I b i j = I I b = 0 (4.22) i i J j J i j J Equation (4.21) co l lapses to a CobbTDouglas production funct ion i f b.. = 0 fo r a l l i and j in which case b^ i s the rec iproca l of the output e l a s t i -c i t y o f the i t ' 1 input and i s s t r i c t l y p o s i t i v e . For the case of Zambian aggregate non-mining sec to r , which i s assumed to employ only two inputs , non-mining cap i t a l K and non-mining Tabor, N, equation (4.21) becomes £n Y = b Q +• b K£riK + b N £nN + ^ b K | < ( £ n K ) + ! 2 ( 4 » 2 3 ) b K ^£nK£nN + j b ^ U n N ) + b^stm where bp = £na. By imposing r e s t r i c t i o n s (4.22) on equation (4.23) and r e -arranging terms we obtain equation (4.24), - 102 -A r 4 } = b 0 + Vn(£} " \ b K N U n | ) 2 + Klnt (4.24) which i s homogeneous of degree 1 in input quan t i t i e s . The marginal procucts of c a p i t a l , $ K , and labor, $ N , can be ea s i l y obtained by p a r t i a l l y d i f f e r e n t i a t i n g equation (4.24) with respect to input quant i t ies K and N re spec t i ve l y : From equation (4.25) and (4.26) i t i s c l ea r that the t rans log product-i o n funct ion contains uneconomic regions over ce r t a in ranges of input space, and i t can therefore f a i l to s a t i s f y the regu l a r i t y condit ions of pro-duction funct ions in a way analogous to the t rans log cost func t i on . For i n -stance with f i n i t e K, $^ becomes negative e i t he r as N -> 0 and b ^ > 0 or as N increases i n d e f i n i t e l y and b ^ < 0 . Therefore f o r equation (4.24) to s a t i s f y the regu l a r i t y condit ions i t must be r e s t r i c t e d with in the economic region which i n the case of a l i n e a r l y homogeneous production funct ion i s character ized by s t r i c t l y po s i t i ve marginal products. Short of imposing b ^ = 0, simultaneous est imat ion o f the production funct ion (4.24) and mar-ginal p roduc t i v i t y condit ions improves the changes of obta in ing po s i t i ve marginal products. Unfortunately equations (4.25) and (4.26) are non- l inear in the unknown parameters. Instead a more convenient way to express the r e -quirement of s t r i c t l y p o s i t i v e marginal products i s to requ i re that the out -put e l a s t i c i t i e s be p o s i t i v e : * K = 3 Y / 3 K = £ [ b K - b K N * n ( £ ) ] (4.25) and * N = 3Y/3N = - b K + b K N £ n ( ^ ) ] (4.26) _' aY K A 3ftnY _ . K 3 K ' Y ' 3£nK D K " b K N *n(^)> 0 (4.27) - 103 -and E N = !i? " Y = IS t 1 " b K + b K N I n ( f ) > 0 < 4 - 2 8 ) While e l a s t i c i t i e s and are var iab les depending on input l e v e l s , they are not observable. We need fu r ther to assume that per fect competit ion pre -v a i l s in the input and output markets. Then the necessary condit ions for p r o f i t maximization are that (a) |Y = P K / P Y and (b) f £ = P^Py (4.29) where P K i s the renta l p r i c e of c a p i t a l , P^ i s the wage rate and Py i s the p r i ce o f output. Conditions (4.29) require that the factors o f production are paid t h e i r marginal products; according to Eu le r ' s theorem under constant returns to s c a l e , i f each input i s paid the amount of i t s marginal product the t o t a l product w i l l be exhausted exact ly by the d i s t r i b u t i v e shares of the input fac tors leav ing zero (excess) p r o f i t s : K f ^ f l + Y (4 .30) By subs t i tu t i ng (4.29) into (4.27) and (4.28) we obtain the d i s t r i b u t i v e share equations: S K - P K ' K / P Y Y = b K " b K N * n ( K / N ) ( 4 > 3 1 ) and; S N =-P.N • N/Py Y = 1 - b K + b K N £ n ( K / N ) (4.32) where i s the r e l a t i v e share o f cap i t a l in the (value of) t o ta l output ( cos t ) , and i s the r e l a t i v e share of labor in to ta l output ( cos t ) . Equa-t ion (4.30) assures that these shares exhaust to ta l output (or co s t ) . C lear -ly d i s t r i b u t i v e shares sum up to one (S^ + = 1). Given appropriate s tochas t i c s p e c i f i c a t i o n and quant i ty , and p r i ce data on inputs and outputs, the production funct ion (4.24) may be estimated - 104 -along with the d i s t r i b u t i v e share equations (4.31) and (4.32) to obtain em-p i r i c a l re su l t s in Chapters 6, and 7 re spec t i ve l y . 4.3 Further Theoret ica l Concepts: Subs t i tu t ion and Pr ice E l a s t i c i t i e s The mining technology to be estimated could be descr ibed by a number of f a m i l i a r concepts such as the A l l en [1938] p a r t i a l e l a s t i c i t i e s of sub-s t i t u t i o n and the p a r t i a l p r i ce e l a s t i c i t i e s of demand for the var iab le i n -puts. Because of the presence of f i xed f a c t o r s , however, a complete des-c r i p t i o n of the technology necess i tates the der iva t ion of new concepts ( e l a s t i c i t i e s ) expressing the re l a t i on sh ip among f i xed inputs , and between the f i xed and var i ab le inputs . In der iv ing these e l a s t i c i t i e s fo r a v a r i -able cost funct ion we fo l low Diewert 's [1974] der i va t ion of analogous e l a s -t i c i t i e s f o r the Var iab le p r o f i t func t ion . Both the conventional and the new e l a s t i c i t i e s can be expressed interms of the f i r s t , second (and cross) p a r t i a l der i va t i ves o f the cost funct ion or a l t e r n a t i v e l y in terms of the f i t t e d shares and the estimated parameters. For a production func t i on , F (V) , A l l en [1938] def ined the p a r t i a l e l a s t i c i t y of subs t i tu t i on between inputs V. and V, as: where F. := 3F/3X-, |F-| nthe bordered Hessian of F and F-. =: the co fac tor Uzawa [1962] has shown that the A l l en e l a s t i c i t i e s of subs t i tu ion be-tween inputs V.j and V h can be def ined in terms of the cost funct ion C(P;Q) as: v ih - (I W W ( l F i h l / l F l ) (4.33) v i h = C * C i h / C i C h i ,h = i ,..., I (4.34) - 105 -Diewert [1974] extended th i s concept to the c lass of va r i ab le p r o f i t func t ions , by der iv ing e l a s t i c i t i e s of transformation between var iab le quan-t i t i e s , inverse e l a s t i c i t i e s o f sub s t i tu t i on between f i xed quant i t i e s and e l a s t i c i t i e s of i n ten s i t y between f i xed and var iab le quan t i t i e s . We de-r i ve analogous e l a s t i c i t i e s fo r the c lass of var i ab le cost funct ions . Con-s idered the fo l lowing uni t va r i ab le cost func t ion : c=.c(P;X) (4.35), where c is the unit cost of product ion, P i s a vector of se rv i ce pr ices of the var i ab le inputs V, and X i s a vector f i xed inputs . Given (4.35) we * * can der ive the fo l lowing e l a s t i c i t i e s at P and X : ( i ) The A l l en p a r t i a l e l a s t i c i t i e s o f subs t i tu t i on between var iab le inputs i and h, are defined as: * * c(P ;X )3 c(P ;X )/3P,.3P * *. 2 - * * ( [ac(P";X")/3P.3 [ a c (P " ;X "J /3P h ] v , h ( P " ; X " ) = — — i f — , i ,h = l,...,I (4.36) C lea r l y i s a normal izat ion of 3V../3P^, such that v ^ h i s i nvar iant to s ca l i ng and equal to v^. . Replacing h by i in equation (4.36) we obtain the own e l a s t i c i t y o f subs t i tu t i on fo r i , v..., and s i m i l a r l y f o r h, i f we replace i by h in (4.36). In the two var iab le input case the own e l a s t i -c i t i e s o f subs t i tu t i on are non-pos i t ive and the cross e l a s t i c i t y non-neg-a t i v e . ( i i ) The inverse e l a s t i c i t y of subs t i tu t i on between two f i xed inputs , j and k i s def ined as, , * * c(P* ; X*)3 2 c(P*;X*)/3X.3X. S i k (P ;X ) = ^ - ^ — — » i j ,k = 1,...,J (4.37) J [3C(P ;X ) /3X j ] [ 3 c (P ;X )/3X k ] As i s a normal izat ion of 3'OK/3X^, i t expresses the percentage change in - 106 -the shadow pr i ce r a t i o w./'oi. of f i xed f ac to r j and k due to a given percent-age change i n the r a t i o o f quant i t i e s X./X. . The own inverse e l a s t i c i t i e s of subs t i tu t i on fo r j , are obtained by rep lac ing k by j in (4.37), and conversely f o r In the two f i xed input case the own inverse e l a s t i c i t -ies o f sub s t i t u t i on are non-negative. ( i i i ) The inverse e l a s t i c i t y o f ' i n t e n s i t y ' between var iab le input i and f i xed input j i s def ined as * * c(P*;X*)3 2c(P*;X*)/3P i9X, i = 1,...,I (4.38) y--(P ;X ) = * * * * 1 J [3c(P ;X )/3P.][3c(P ;X 3 = 1.....J Thus y j^ i s a normal izat ion of 3V^/3Xj, the change in the i t h var i ab le i n -put demand with respect to a change in the j f i xed f ac to r quant i ty , with the normal izat ion chosen so that y^ i s invar iant to s c a l i n g . The signs of y.. and y.. cannot be determined a p r i o r i . For the t rans log var i ab le cost funct ion given by equation (4.12) the above e l a s t i c i t i e s can be defined in terms of the parameter estimates a ^ * * * * and a^ R and the f i t t e d shares S^, S^, and S R where and are def ined as in equations (4.13) and (4.14) whi le S z = 3£nC/3£nZ and S R = 3£nC/3£nR . (To economize on notat ion henceforth we w i l l be using a f o r the e l a s t i c i t i e s o f subs t i tu t i on and i n ten s i t y and n f o r the p r i ce e l a s t i c i t i e s ) . The A l l en p a r t i a l e l a s t i c i t i e s o f sub s t i tu t i on in the case of the t rans log are ob ta in -ed as: 0ML = ( S M S L + a M L ) / S M S L ( 4 ' 3 9 ) °MM = ( S M 2 " S M " a M L ) / S M 2 < 4 - 4 0 ) and s i m i l a r l y fo r . The inverse e l a s t i c i t i e s of sub s t i tu t i on are ob-ta ined as: - 107 -( i i ) ' o 1 R = (S*z • S* + a Z R ) / S * S* (4.41) h i = {s*i ~ Sl ~ a Z R ) / S Z 2 ( 4 ' 4 2 ) and s i m i l a r l y f o r a R R . F i n a l l y the e l a s t i c i t i e s of i n t e n s i t y are obta in -ed as: (111)' a M Z = (S* s j - a Z M ) / S * S* (4.43) and s i m i l a r l y for a ^ z , a M R and a ^ R . The p a r t i a l p r i ce e l a s t i c i t i e s o f demand fo l low d i r e c t l y from the A l l e n p a r t i a l e l a s t i c i t i e s o f sub s t i t u i on : nML = ( 3 M / 8 P L ) (P L/M) = 8£nM/3£nP L = S* • a M ( _ ( 4 4 4 ) nMM = ( P M / M ) = 9 * n M / 9 * n P M " S M °MM (4.45) and s i m i l a r l y f o r nLM and. • The inverse p r i ce e l a s t i c i t i e s f o r f i xed inputs are obtained d i r e c t l y from the inverse e l a s t i c i t i e s o f s u b s t i t u t i o n : nZR = (3fcl-z/3R) (R/W2) = 3£nM z /3AnR = S* a Z R (4.46) n ZZ = (8<VsZ) ( Z / W Z ^ = 3£nloz/35,nZ = S z a z z (4.47) and s i m i l a r l y f o r nRZ and nRR . Turning to the non-mining technology, we need only to der ive the A l l en p a r t i a l e l a s t i c i t i e s o f sub s t i t u t i on between non-mining c a p i t a l K ahd non-mining labor N. From formula (4.33) we have _ -T 1 h X i X h |F F l h ' i , h = K,N (4.48) where - ior-F K TKK F KN F N F KN F NN i s the bordered Hessian of the production function F, | F . J i s the co factor of 3 2 F / 9 X i 3 X h in |F| and f. == 3F/3X. . In terms of the f i t t e d d i s t r i bu t i ve shares and and the para-meters estimate a ^ , of. the translog production formula (4.24) the A l len cross pa r t i a l e l a s t i c i t y of subst i tut ion N defined as: * * * JN 1 U K N " " K an own e l a s t i c i t y of subst i tut ion for K as (4.49) CTKK * K K *2 A K N ) / S K (4.50) and s i m i l a r l y for - 109 -FOOTNOTES TO CHAPTER 4 1. The d i s t i n c t i o n between f i x e d and var iab le inputs though not pecu l i a r to mining i s f a r more ser ious in l o c a t i o n - s p e c i f i c i n d u s t r i e s , such as mining. It becomes p a r t i c u l a r l y important in the case of fore ign min-ing companies operat ing in developing countr ies because of the ever--present p o s s i b i l i t y o f na t i ona l i z a t i on or appropr iat ion without ade-quate compensation. 2. F i x i t y o f l oca t ion and varying qua l i t y o f inputs i s c h a r a c t e r i s t i c of a l l resource-based indus t r ie s but i s of p a r t i c u l a r importance in mining and f o r e s t r y . F i x i t y of quant i ty i s a c h a r a c t e r i s t i c s p e c i f i c to non-renewable resources and in p a r t i c u l a r to o i l and minera ls . 3. A grade s l i g h t l y higher than the copper content of the common rock i s employed here as the c u t - o f f grade that d i s t ingu i shes ore from common rock. Foreaadeta i led ana lys i s o f th i s point see sect ion 5.5. 4. Soott [1967: 46]. 5. Dasgupta and Heal [1974: 10] suggested the i nc lu s i on of the resource stock in the cost funct ion to i nd i ca te that "one i s , as i f i t were, digging deeper when the stock has shrunk". 5a. See sect ion 5.5 fo r a prec i se d e f i n i t i o n of the average grade of cumu-l a t i v e product ion. 6. Banks [1974: 83] noted that the "presence of a kind of Horndal or l e a r n -ing by doing e f f e c t " in Zambian copper mining. 7. Rapping [1965] and Sheshinski [1967] a lso represented experience or " l ea rn ing by doing" by cumulative output. Rapping using a l o g - l i n e a r production funct ion f o r U.S. shipping yards found that cumulative out-put was super ior to calendar time as explanatory var iab le of technica l change. 8. We draw mainly from Diewert 's [1974] expos i t ion of the dua l i t y theory. 9. Ib id. pp. 133 - 146. 9S. The flows of serv ices are assumed proport ional to the stocks. 10. This assumption implies that the rate of copper ext rac t ion ( leve l o f annual output) i s not a choice var i ab le of the copper industry, in terms of our intertemporal model output Q i s preassigned to the cop-per industry by the host government. This contro l over output was sought by the Zambian government a f t e r independence through the export tax and s ince 1970 by the 51 percent na t i ona l i z a t i on of the copper i n -dustry. - no -11. We assume that the cost minimization takes place at the industry l e v e l . Mining indus t r ie s in developing countr ies usual ly cons i s t of one m u l t i -nat ional company ! vor, as in the case of Zambia, o f two i n t e r r e l a t e d a f f -i l i a t e s o f mul t inat iona l corporat ions . 12. Mining costs include a l l va r i ab le costs of production incurred from the stage of ext rac t ion to the stage of export. 13. See Diewert [1971]. 13a. I am indebted to Professor Diewert fo r provid ing me with these proofs. 14. See Shephard [1970] and Diewert [1974]. 15. Other members of the c lass o f f l e x i b l e funct iona l forms are the Gener-a l i z e d Leont ie f (GL), the Square Rooted Quadratic (SRQ) and the Gener-a l i z e d Cobb-Douglas (GCD). The t rans log funct iona l form (TL) was chosen over these a l t e r n a t i v e forms f o r a number of reasons: ( i ) TL i s preferab le to GCD which su f fer s from lack of invar iance to s ca l i ng as shown by Wales and Woodland [1977]; ( i i ) TL performs genera l ly bet -t e r than SRQ (see Khaled [1977]); and ( i i i ) TL conveniently reduces to a Cobb-Douglas form i f a ^ = 0 fo r a l l i and h (see equation 4.3 beilow). Recently Khaled [1977] developed the General ized Box-Cox (GBC) funct iona l form which y i e l d s the above funct iona l forms as a spec ia l parametric case thus provid ing a nested framework f o r d i s c r im ina t ing among them. The GBC approach was not used i n th i s study but both the SQR and the GL were attempted. However, the resu l t s were less s a t i s -f ac tory than those of the trans log form. 16. The var i ab le cost funct ion i s a spec ia l case of the va r i ab le p r o f i t funct ion (see Diewert [1974]). Burgess [1976] was the f i r s t to use a t rans log j o i n t cost func t i on . See a l so Christensen and Greene [1976]. 16a. For a d e f i n i t i o n ' o f constant returns to var i ab le inputs and a j u s t i f i c -a t ion fo r th i s assumption see footnote 16a to Chapter 3, and Appendix B. 17. Woodland [1976: 20]. 17a. The purpose of th i s paragraph i s to provide ins ights into the i n t e r p r e -ta t i on of the s t a t i c shadow pr ices below. 18. The theo re t i c a l basis f o r incorporat ing exogenous neutral technica l change into a cost production funct ion i s r e l a t i v e l y weak. It may be introduced by t rea t ing calendar time j u s t as another f i x e d input or simply as a trend va r i ab le . (Other representat ions of technica l change may a l so be dev i sed) . Here we t rea t time as a tcend va r i ab l e . The time var i ab le may enter e i t h e r as T or as £nx. Jorgenson [1974] uses x whi le Appelbaum and Harr i s [19*74] use am. However, t h e i r treatment o f technica l change i s more general than ours as they enter e T or x as an add i t iona l f i xed input. S im i l a r i s the treatment of f ac to r augment-a t i n g techn ica l change by Berndt and Wood [1975]. In t h e i r formulat ion - m -Hicks-neutra l techn ica l change can be tested as a spec ia l case. Consider the i n te rp re ta t i on and j u s t i f i c a t i o n given in footnote 16a o f Chapter 3, and i n the Appendix B. We fo l low a d i f f e r e n t approach f o r the non-mining sector f o r two good reasons: ( i ) data a v a i l a b i l i t y ; and ( i i ) consistency with the s t r u c t -ure of the planning model o f Chapter 3. Of course, one of the d i s t r i b u t i v e share equations must be dropped as i t i s not independent of the other, given + S, = 1 . CHAPTER 5 DATA: REQUIREMENTS SOURCES AND METHODOLOGY The ob jec t i ve of the present chapter i s to descr ibe the data r equ i re -ments of the est imating equations, the data sources, and the methodology of construct ing the required t ime-ser ies when such ser ies were not a v a i l a b l e . The econometric est imation of the two production technologies (mining and non-mining), introduced in the preceeding chapter requires long t ime-ser ies on the respect ive pr ices and quant i t i e s of inputs and outputs. Most of these ser ies were e i the r unavai lab le at the required leve l of aggregation (e.g. output and cap i t a l stocks) or completely unavai lab le (e.g. rental pr ices and copper resource s tock) . Thus, i t became necessary to employ standard methods such as the perpetual inventory procedure of construct ing cap i ta l stock ser ies and the D i v i s i a Index method of aggregation. In some cases the ex i s t i ng metho-dology needed to be modif ied or extended before the appropriate ser ies could be constructed. Each of the f i v e sect ions of th i s chapter is organized in the fo l lowing pat tern: f i r s t , the need fo r some ser ies i s e s tab l i shed ; then, ex i s t i ng re levant ser ies are reviewed and i f inappropr iate a method f o r construct ing the required ser ie s i s introduced; and l a s t , the Zambian sources of the -'raw' data required by the methods are discussed and the constructed ser ies are repor ted . 1 Most of the data processing f o r th i s study uses the TSP (Time Series Processor) program but some FORTRAN programming was a l so requ i red . The computations are c a r r i e d out on an IBM 370-168 computer at the Un iver s i t y of B r i t i s h Columbia -113-5.1 Construct ion of Capita l Stock Ser ies : The Perpetual Inventory Approach The est imation of the aggregate production funct ion fo r the non-mining sector introduced requires a time ser ies on non-mining cap i t a l stock. S i m i l a r l y , the est imation of the va r i ab le cost funct ion f o r the mining sector requires data on f i xed and va r i ab le c a p i t a l . There have been two e a r l i e r attempts to compute aggregate or sectora l cap i t a l stock f i gures f o r Zambia. The Central S t a t i s t i c a l O f f i c e (C.S'.O) of Zambia, computed 1971 estimates of cap i t a l stock in Mining and Quarrying (K710.7 mn), Manufacturing (K184.2 mn) and Construct ion (K55.4 mn), 2 using the 'market va luat ion approach' . McPherson [1976a] estimated cap i t a l stock ser ie s f o r the period 1945-72 f o r the aggregate economy and 3 s ix ind iv idua l sectors using the 'perpetual inventory approach' explained below. We do not use McPherson's ser ies f o r two reasons. F i r s t , fo l lowing Jorgenson and G r i l i c h e s [1967] and Woodland [1972b] we prefer 4 a geometric deprec ia t ion rate to the actual c ap i t a l consumption 5 allowances (used by McPherson). Second, f o r our purposes, the cap i t a l stocks of a l l the non-mining sectors must be aggregated into one s e r i e s , and the mining cap i t a l stock disaggregated into f i xed and va r i ab le cap i ta l . The construct ion of sectora l stocks i s discussed in th i s sect ion whi le problems of aggregation are dea l t with in sect ions 5.3 and 5.4. Using the standard perpetual inventory approach with a constant rate of d e p r e c i a t i o n 7 we construct cap i t a l stock ser ies f o r the f i v e non-mining sectors : commercial a g r i c u l t u r e , manufacturing, transport and communications, construct ion and serv ices f o r the period 1945-72. Using the same approach we construct f i gures on f i xed cap i t a l (bu i ld ing -114-and engineering s t ructures ) and va r i ab le cap i t a l (machinery and equipment) f o r the mining sec tor . The perpetual inventory procedure involves the cumulation of net f i xed investment commencing with some benchmark cap i t a l stock f i g u r e . The cap i t a l stock a t the beginning of period t , K t i s equal to the cap i t a l stock a t the beginning of the preceeding period 1 plus the net rea l investment during the period t - 1 , 1^  ^, i . e . : K t = I (1-6)* I (5.1) or a l t e r n a t i v e l y K t = ( 1 - 6 ) ^ + lt_^ (5.2) where s i s a constant one-period deprec ia t ion ra te . Unl ike the conventional formula , = (1-6) « t ^ + 1^, which measures the cap i t a l stock at the end of each period formula (5.2) measures the cap i t a l stock at the beginning of each period assuming that investment made during the current period does not become productive u n t i l the beginning of the next per iod. The above procedure requires ( i ) a time se r ie s on rea l gross f i xed cap i t a l format ion, ( i i ) a benchmark observat ion on cap i t a l stocks and ( i i i ) ' a constant rate of deprec ia t ion (or the average l i f e of c a p i t a l s t ructures and equipment). The gross f i xed investment ser ies in 1967 pr ices fo r the f i v e non-mining sectors are obtained from McPherson [1976a: Table Z1.3] and reproduced in Table XXXII. The o r i g i na l source is the National Accounts [1954, 1964, 1971] 9 which g ive the investment se r ie s in current pr ices and the corresponding pr i ce d e f l a t o r s . The fo l lowing deprec ia t ion rates and 1945 benchmark c a p i t a l stocks were employed: -115-Depreciat ion Rate (PercentJ Benchmark (Kmn 1967 pr ices ) Sector Commercial Ag r i cu l tu re 10 1.300 Manufacturing 8 1.300 Transport and Communica-t ion 5 22.300 Construct ion 20 0.600 Services 2.1 23.800 The above f i gures were taken from McPherson [1976a]. As noted e a r l i e r McPherson did not use constant geometric deprec iat ion rates ( in percent) in construct ing his cap i t a l s e r i e s . However, he did use one-period deprec ia t ion rates to der ive the above benchmark s tocks 1 ^ from the 1945 cap i t a l consumption allowances ( in Kwachas). Table V below reports the constructed cap i t a l stock ser ies fo r the f i v e non-mining subsectors. In comparison to McPherson's estimates our f i gures are somewhat higher f o r manufacturing, lower f o r t ransport , construct ion and serv ices and the same for a g r i c u l t u r e . 1 1 For the mining sector we d i s t i n gu i sh between two types of c a p i t a l : ( i ) machinery, equipment, and other imported mater ia l s (henceforth machinery) f o r operat ing the mines and ( i i ) . b u i l d i n g and engineering s t ructures (henceforth s t ructures ) f o r mine development. Machinery i s assumed to have high resa le va lue, i t can be rented out, and var ied 12 f r e e l y in the shor t - run . S t ructures , such as mines, smelters, and m i l l s are assumed to have low or zero resa le va lue, cannot be rented out, or var ied f r e e l y in the shor t - run, and are thus treated as f i xed inputs. Of course, some expansionary investment in new mines and s t ructures i s -116-TABLE V ESTIMATED CAPITAL STOCKS FOR NON-MINING SECTORS, ZAMBIA, 1945-1973. (M i l l i ons of Kwachas, 1967 p r i c e s ) . YEAR KAG KMF KTR KCN KSV 1945 1.3000 1.3000 22.3000 0. 6000 23.8000 1946 2.1542 2.0093 21.7966 1. 2121 25.8714 1947 3.1476 2.8816 21.7019 1. 9007 28.4335 1948 3.9973 3.5334 21.9137 2. 3253 32.6416 1949 5.3043 4.6685 22.1401 3. 1433 41.5481 1950 6.6588 5.8035 23.1836 3. 8786 55.3276 1951 8.1801 7.0786 24.3297 4. 6783 72.5111 1952 9.5378 8.1020 25.1015 5. 1810 89.3931 1953 11.4041 9.5426 26.2872 7. 0321 112.0050 1954 13.2988 10.9519 27.6155 7. 5916 142.5980 1955 16.3485 13.7231 38.5695 9. 3766 192.6280 1956 17.8117 20.0658 51.6410 10. 1035 242.8070 1957 19.3201 26.3589 64.9191 10. 8714 295.2270 1958 21.0425 30.3192 85.4640 9. 8351 358.4130 1959 21.8580 30.9281 97.0598 9. 8738 411.5670 1960 24.0480 33.0057 106.0170 10. 4486 464.6920 1961 24.9317 35.8298 114.5360 9. 1708 500.5040 1962 25.3582 36.6092 121.7360 8. 6594 529.8050 1963 25.0142 35.8052 129.1750 7. 7140 557.1980 1964 27.6192 35.9782 131.5370 6. 4200 578.9300 1965 31.0607 39.4753 131.7240 6. 3350 608.7040 1966 38.5294 46.9437 136.5570 15. 9129 663.1580 1967 46.2780 59.3933 170.1880 22. 4410 728.2960 1968 57.8502 72.6418 196.1790 34. 6670 816.4300 1969 70.7808 104.9640 201.4560 43. 9336 910.8500 1970 82.3193 117.7060 202.5900 52. 0427 985.6030 1971 87.7068 141.8160 210.5370 65. 1163 1076.4800 1972 98.1579 147.7970 233.3950 63. 4995 1167.7900 1973 102.0300 171.9380 250.3180 60. 1446 1255.7800 KAG: Commercial Ag r i cu l tu re KMF: Manufacturing KTR: Transport and Communications KCN: Construct ion KSV: Services Source Estimated using formula (5.2) and data on real gross f i xed investment (Table XXXII) -117-taking place but i t i s ne i ther a major f r a c t i o n of the t o t a l , nor a part of the short-run ca l cu l a t i on s of the va r i ab l e - co s t minimizing f i rm. Indeed "the major i ty of cap i t a l expenditures are needed in order to maintain the 13 leve l of mine capac i t i e s due to dep let ion (and) r econs t ruc t i on . " This suggests the fu r ther d i s t i n c t i o n between investment in new mine develop-ment, and investment in maintainance or marginal extensions of ex i s t i ng mines. While the l a t t e r usua l ly becomes e f f e c t i v e each year , investment in developing new mines does not become e f f e c t i v e u n t i l the mine i s completed. Thus, in construct ing a stock ser ies f o r s t ructures we accumulate a l l expansionary investment to the opening date of the new mine. The National Accounts [1954, 1964, 1971] report separate f i gures f o r gross investment in (a) machinery and equipment (b) mine development, bu i ld ings and works fo r the years 1954-1969, and aggregate f i gures fo r the years 1945-53. For th i s e a r l i e r period during which no new mine development was taking place we broke down the aggregate f i gures according to the a l l o c a t i o n of 1954, which was a l so a year of non-expansionary 15 investment. The 1970 f i gure was obtained from Bostock and Harvey [1972] and was d iv ided between machinery and s t ructures according to the 1969 a l l o c a t i o n . The r e s u l t i n g gross investment se r ie s in machinery and s t ructures are reported in Table XXXIV. The ser ies on s t ructures was fu r ther adjusted f o r new mine development. In the case of Zambia, the bulk of investment in new mine development took place during 1927-39 when the four major mines were constructed. Since then, while several extensions were gradual ly made to ex i s t i ng mines, only three rather minor mines were added. The medium s i ze Bancroft mine -118-opened in 1957 and had a major expansion completed in 1 9 5 9 . 1 7 Two smaller mines, Chibuluma and Chambishi opened in 1956 and 1965 r e s p e c t i v e l y . The bulk of development investment f o r Bancroft and Chibuluma took place during 1955-57 and was a l l oca ted between the two mines as fo l lows: we f i r s t , assumed that the non-expansionary development investment continued at i t s 1954 leve l f o r the years 1955-57. Then, we add to the 1956 non-expansionary investment Kl2 mn which Coleman [1971:155] reports as being the o r i g i na l cost of developing the mine. The remaining balance of K29.7 mn from the 1955-57 new mine development expenditure i s a t t r i bu ted to Bancroft Mine and a l l oca ted between 1957 and 1959 according to the 1 g expansion completed in each year. The investment in developing the Chambishi mine was spread over the period 1961-64. Assuming that the non-expansionary investment continued at i t s 1960 leve l we accumulated the remaining balance to 1965 when the Chambishi mine commenced product ion. The adjusted ser ies of investment in s t ructures i s reported in Table XXXIV. The gross investment f i gures in machinery was then def la ted by the import p r i ce index (1967 = 100) of mining machinery and equipment. For the years 1954-70 th i s index was obtained from McPherson [1976a: Table 1114), while f o r the years 1945-53 i t was obtained by extrapo lat ing the 1954 f i gu re at the same rate of change as the index of un i t value of imports, a l so obtained from McPherson [1976a: Table NZ1.2). To obtain a p r i ce d e f l a t o r f o r investment in mining s t ructures we assumed that the aggregate mining investment de f l a t o r given by McPherson [1976a: Table Z1.5] i s a weighted average of the ind iv idua l ind ices fo r s t ructures and machinery with the weights being the respect ive investment quan t i t i e s . Then, the p r i ce d e f l a t o r f o r s t ruc tu re s , X 0 , was obtained as: -119-2 X, = (X - W,X.)/W9 with. E W. - 1 (5.3) where X i s the aggregate p r i ce index, X-j i s the p r i ce index f o r machinery, and W.j, are the respect ive shares of machinery and s t ructures in to ta l investment. These three p r i ce ind ices are reported in Table XXXV. The benchmark stocks f o r s t ructures and machinery were obtained by assuming that the real deprec ia t ion allowances during 1945 represented f i xed proport ions (.07 f o r s t ructures and .10 f o r machinery) of the actual 19 c a p i t a l stocks a t the beginning of 1945. Unfortunately we could only 20 obtain one aggregate deprec ia t ion allowance f i gu re f o r the en t i r e mining stock of which we a t t r i bu ted 38 percent to machinery and 62 percent to 21 s t ruc tures . Under these assumptions the fo l lowing s t a r t i n g values were der ived: K90.200 mn f o r s t ructures and K28.500 mn fo r machinery, both in 22 1967 p r i ce s . Employing the perpetual inventory method explained e a r l i e r in t h i s s e c t i o n , and assuming deprec ia t ion rates of .07 f o r s t ructures and .10 f o r machinery, we computed the time ser ies f o r f i xed and va r i ab le cap i t a l reported in Table VI below. A l t e r n a t i v e ser ies with lower deprec ia -t ion rates and higher s t a r t i n g values were a lso constructed and are reported in the same Table. Having constructed the required stock ser ies we now proceed to const ruct t h e i r renta l or serv ice p r i ce s . 5.2 Construct ion of Rental P r i ce s : The Jorgenson-Gr i l i ches -D iewert Procedure Rental pr i ces are required f o r the aggregation of the f i v e non-mining cap i t a l stocks and est imation of the production func t i on . S i m i l a r l y , in the case of mining, the renta l p r i ce of machinery i s needed f o r the aggregation of imported va r i ab le inputs , (machinery and expatr ia te t e c h n i c -120-TABLE VI ESTIMATED STOCKS OF FIXED AND VARIABLE CAPITAL IN MINING AND CORRESPONDING FLOWS OF REPLACEMENT AND NET INVESTMENT, ZAMBIA, 1945-1970. (M i l l i ons of Kwachas, 1967 p r i c e s ) . YEAR KVCl NIVl REVl KVC2 NIV2 REV2 1945 28.5000 1.2178 2.8500 35.6000 1 .2198 2 .8480 1946 28.3238 -0.1586 2.8324 35.4258 -0 .1602 2 .8341 1947 29.8582 1.3810 2.9858 36.9585 1 .4101 2 .9567 1948 34.9637 4.5949 3.4964 42.0931 4 .7238 3 .3675 1949 41.6299 5.9996 4.1630 48.8882 6 .2515 3 .9111 1950 52.2676 9.5740 5.2268 59.7779 10 .0185 4 .7823 1951 62.9500 9.6141 6.2950 70.9048 10 .2367 5 .6724 1952 76.5046 12.1992 7.6505 85.0820 13 .0431 6 .8066 1953 90.3595 12.4694 9.0359 99.7808 13 .5229 7 .9825 1954 112.8170 20.2118 11.2817 j:33.2920 21 .6301 9 .8634 1955 127.9870 13.6529 12.7987 139.8800 15 .2612 11 .1904 1956 143.4930 13.9555 14.3493 156.9940 15 .7453 12 .5596 1957 162.2240 16.8580 16.2224 177.5150 18 .8792 14 .2012 1958 168.4800 5.6304 16.8480 185.?7920 7 .6149 14 .8634 1959 171.1810 2.4308 17.1181 190.4780 4 .3106 15 .2382 1960 176.4770 4.7661 17.6476 197.6530 6 .6015 15 .8123 1961 187.5150 9.9348 18.7515 210.5270 11 .8441 16 .8422 1962 193.1670 5.0865 19.3167 218.0880 6 .9561 17.4471 1963 192.4150 -0.677T0. 19.2414 219.2060 1 .0279 17 .5364 1964 190.2460 -1.9515 19.0246 218.7420 -0 .4262 17 .4994 1965 190.5080 0.2358 19.0508 220.5300 1 .6443 17 .6424 1966 185.2290 -4.75101 18.5229 216.6590 -3 .5609 17 .3327 1967 179.2060 -5.4206 17.9206 211.8260 -4 .4461 16 .9461 1968 165.9980 -•11 .8873 16.5998 199.5930 -11 .2549 15 .9674 1969 173.9440 7.1511 17.3944 208.1710 7 .8918 16 .6537 1970 170.8220 -2.8093 17.0822 205.7900 -2 .1903 16 .4632 KVCl Stock of Var iab le Capita l i . e . Machinery and Equipment (6 = NIVl Net Investment (<s = .10) REV2 Replacement Investment (6 = . 10) KVC2 Stock of Var iab le Capi ta l (6 = .08) NIV2 Net Investment (6 = .08) REV2 Replacement Investment (6 = . 08) 6 Oner-period-: d constant deprec iat ion rate (continued) -121-TABLE VI (continued) YEAR KFC3 NIF3 REF3 KFC4 NIF4 REF4 1945 90 .2000 -4 . 0077 6. 31400 126. 3000 -4. 0087 6. 3150 1946 85 .3730 -4 . 4891 5. 97611 121. 4720 -4. 5866 6. 0736 1947 82 .1824 -2 . 9673 5. 7528 118. 1840 -3 . 1237 5. 9092 1948 81 .6652 - 0 . 4810 5. 7166 117. 5100 -0 . 6399 5. 8755 1949 83 .7347 1. 9247 5. 8614 119. 4210 1. 8151 5. 9710 1950 87 .2785 3. 2957 6. 1095 122. 8550 3. 2624 6. 1428 1951 91 .2501 3. 6936 6. 3875 126. 7930 3. 7414 6. 3397 1952 97 .4412 5. 7577 6. 8209 133. 0320 5. 9270 6. 6516 1953 105 .0950 7. 1184 7. 3567 140. 8560 7. 4322 7. 0428 1954 117 .8480 11. 8599 8. 2494 153. 9220 12. 4132 7. 6961 1955 124 .8200 6. 4843 8. 7374 161. 4480 7. 1493 8. 0724 1956 150 .4540 23. 8392 10. 5318 187. 7460 24. 9836 9. 3873 1957 172 .5780 20. 5752 12. 0804 211. 0150 22. 1049 10. 5507 1958 180 .5440 7. 4089 12. 6381 220. 5110 9. 0214 11. 0255 1959 211 .4520 28. 7447 14. 8017 253. 0320 30. 8947 12. 6516 1960 211 .7200 0. 2493 14. 8204 255. 4500 2. 2972 12. 7725 1961 212 .0170 0. 2758 14. 8412 257. 7940 2. 2272 12. 8897 1962 211 .5140 - 0 . 4681 14. 8060 259. 2420 1. 3758 12. 9621 1963 211 .1240 - 0 . 3628 14. 7786 260. 6960 1. 3811 13. 0348 1964 208 .9530 -2. 0182 14. 6267 260. 2700 -0 . 4049 13. 0135 1965 218 .1570 8. 5597 15. 2710 271. 0870 10. 2764 13. 5543 1966 216 .4970 - 1 . 5446 15. 1548 271. 1420 0. 0531 13. 5571 1967 223 .1420 6. 1801 15. 6199 279. 3850 7. 8307 13. 9692 1968 233 .1450 9. 3027 16. 3201 291. 0390 11. 0709 14. 5519 1969 251 .9360 17. 4758 17. 6355 311. 5980 19. 5314 15. 5799 1970 255 .8900 3. 6779 17. 9123 317. 6080 5. 7098 15. 8804 KFC3: Stock of F ixed Capi ta l i . e . Bu i ld ing and Engineering Structures (6 = .07) NIF3: Net Investment (6 = .07) REF3: Replacement Investment (6 = .07) KEF4: Stock of Frixed Capita l (6 = .05) NIF4: Net Investment (6 = .05) REF4: Replacement Investment (6 = .05) 6 : One-period constant deprec iat ion rate Source: Estimated using formula (5.2) and data on gross f i xed i nves t -ment (Table XXXIV) and investment def la tor s (Table XXXV). -122-c i a n s ) , and the est imat ion of the va r i ab le cos t func t i on . The purchase p r i c e of investment goods, whi le appropriate fo r the aggregation of investment goods output, cannot be used f o r the aggregation of cap i ta l 23 se rv i ce s . The corresponding serv ice p r i ce or user cost of cap i t a l should be used. However, given the lack of markets fo r cap i t a l serv ices in general there i s no observable p r i ce index r e f l e c t i n g the user cost of c a p i t a l . To der ive such an index we employ a procedure i n i t i a t e d by Jorgenson and G r i l i c h e s [1967] and discussed ' further by Diewert [1972] [1977] and Woodland [1972b]. There have been no previous attempts to construct renta l pr ices of c ap i t a l or other durables f o r Zambia. The Jorgenson-Gr i l i ches-D iewert procedure der ives the renta l p r i ce of c ap i t a l goods, Q .^, by supposing the existence of competit ive ' l e a s i n g ' f irms which equate the purchase cost of a un i t of cap i t a l good, Q^, to i t s 24 depreciated value in the next period (1 - 6) Q t + 1 discounted to period t, plus the net renta l income from period t , P^-T: (1--5) Q t + 1 l + r t where, T = u t ( W Q t - W + ¥ t ( 5 - 5 ) i s the sum of corporate and property taxes, P^ i s the renta l p r i ce of c a p i t a l , i s the e f f e c t i v e corporate tax r a t e , X^ . i s the property tax r a te , 6 i s one period combined deprec ia t ion and obsolescence ra te , i s the proport ion of deprec ia t ion a l lowable f o r taxat ion purposes and r^ . i s the nominal i n t e r e s t or d iscount rate app l i cab le to period t . By subs t i tu t ing (5.5) in to (5.4) and so lv ing f o r P^ we obtain the renta l p r i ce formula: -123-r t Q t + 6 Q T + 1 - (0 t+1 - Qt) - 0 + r t).U tV t<SQ t + .(1 + r t ) ( l + r t ) ( l - U t ) X t Q t Equation (5.6) i s the d i s c re te time formula derived by Diewert [1977] and i s s im i l a r to the continuous time formula der ived by Jorgenson and G r i l i c h e s [1967], except f o r the term (1 + r) in the denominator of (5.6). Woodland [1972b] [1977] employed a vers ion of (5.6) s im i l a r to the one developed by Jorgenson e t . a l . In what fo l l ows , we use Woodland's formula assuming s t a t i c expectations with respect to pr ices of cap i t a l goods, Qt+^ = 0^, thus, se t t ing expected cap i t a l gains equal to zero. The property tax term i s a l so set equal to zero s ince such taxes have been unimportant in the case of Zambia. Then Woodland's formula becomes: [ r . + SO-U.V. ) ] Q P. = i = 1 (5.7) ] - U t where U i s a weighted average of the corporat ion tax rate and the personal 25 tax r a te . The renta l p r i ce formula (5.7) involves an i n te re s t r a te , r^. While i t i s genera l ly agreed that r^ should be a nominal rate to take account of i n f l a t i o n and should r e f l e c t the marginal cost of borrowing f o r net borrowers or the marginal return from lending fo r net lenders , in p rac t i ce two d i f f e r e n t rates are used. Woodland [1972b] [1975], and Jorgenson and G r i l i c h e s [1967] employed an interna l or i m p l i c i t rate of r e t u r n , while Woodland [1977] uses "an exogenous bond rate which may or may not apply to the f i rm under con s i de ra t i on . " The present study uses both approaches: the expected ( i m p l i c i t ) rate of return i s used fo r a l l -124-sectors except mining f o r which an exogenous rate is used. The two approaches are discussed in the above order. The expected ra te of return to c a p i t a l , r^, in a world of s t a t i c 27 expectat ions, could be def ined as: B, - Q tK - U (B - V SQ K ) r. = -5 L _ J 1 t t ( 5 > 8 ) V t where i s the sectora l cap i t a l stock and B t i s the rental income of the sector which i s obtained by assuming that labour and cap i t a l earnings completely exhaust the f a c to r payments B t = Y t - W tL t (5.9) where• i s the value added, are the average annual earnings of labor and L t i s sectora l l abor . F i n a l l y , V^Q tK i s assumed equal to the actual 28 cap i t a l consumption allowances in the sector . Once the i m p l i c i t rate of return i s computed according to (5.8) i t can be subst i tuted in (5.7) to der ive the renta l s or serv ice pr ices fo r the sectora l cap i t a l s tocks. The use of formulae (5.8) and (5.7) in construct ing the rental pr ices of the f i v e non-mining cap i t a l stocks fo r the years 1954-72, requires the fo l lowing data f o r each sector : ( i ) purchase pr i ce index f o r c ap i t a l equipment; ( i i ) actual c ap i t a l consumption allowances ( in Kwachas); ( i i i ) value added and the wage b i l l ; and ( iv ) deprec ia t ion ra tes . The corporat ion and personal tax rates are a l so requ i red. The purchase p r i ce ind ices are taken to be the gross investment de f l a to r s (with 1967 as base 29 year) given in Table XXXIII. The cap i t a l consumption al lowances, reported in Table XXXVI were obtained by r e f l a t i n g the real ser ies given in McPherson [1976a: Table Z1.4]. The value added, average annual -125-earnings, and the labor force f o r each sector are discussed in the:.next sect ion and are reported in Tables XLI, XLIV and XLIII r e spec t i ve l y . We use sectora l deprec ia t ion rates i den t i c a l to those used f o r the construct ion of cap i ta l stocks in the proceeding sec t ion . Due to the lack of separate data on corporat ion and personal taxes we use the r a t i o of d i r e c t taxes 30 to non-subsistence Net National Product as the re levent tax r a te . Data on d i r e c t taxes were obtained from National Accounts [1964: Tables 8, and 9] and various issues of Monthly Digest of S t a t i s t i c s , of C.S.O. A l l data employed in the der i va t i on of the tax rate appear in Table XXXVII. The computed renta l s or serv ice pr ices fo r the f i v e non-mining sectors are reported in Table VII of th i s -Chapter . For the mining sector only the renta l formula (5.7) need be used s ince an exogenous ra te of i n t e r e s t i s chosen over the widely f l u c t u a t i n g 31 i m p l i c i t rate of r e t u r n . The annual average y i e l d (percent) on U.S. corporate bonds ('Baa c l a s s i f i c a t i o n ' in Moody's r a t i n g ) , sca led up by 32 33 a f a c to r of two was taken as the re levant i n te re s t r a te . This bond rate was obtained from the December issues 1945-1971 of the Federal Reserve B u l l e t i n and i s reproduced in Table XXXVIII. Four a l t e r n a t i v e renta l pr ices f o r va r i ab le cap i t a l (machinery and equipment) were computed by assuming d i f f e r e n t deprec ia t ion and tax ra tes . Two deprec ia t ion rates were used, .10 and 0.08, i den t i c a l to those used fo r the const ruct ion of the stock of machinery. P r i o r to 1970, there were no mining p r o f i t taxes as such. There were, always, mineral r o y a l t i e s , and s ince 1966, export taxes, but the only d i r e c t tax was a general 45 percent income tax on -126-TABLE VII COMPUTED RENTAL PRICES B0R NON-MINING CAPITAL STOCKS, ZAMBIA, 1954-1972. YEAR RPAG RPMF RPTR RPCN RPSV 1954 0.0994 0.2664 0.1235 . 0.2890 0.0938 1955 0.0426 0.2818 0.1241 0.3097 0.1183 1956 0.1768 0.2792 0.1220 0.3434 0.1118 1957 0.2433 0.2884 0.1123 0.4698 0.0953 1958 0.0650 0.2358 0.0674 0.4130 0.0809 1959 0.1793 0.1832 0.07981 0.3641 0.0814 1960 0.1336 0.1932 0.1008 0.4663 0.0830 1961 0.2473 0.2966 0.0865 0.4266 0.0801 1962 0.2325 0.2348 0.0772 0.6008 0.0804 1963 0.1646 0.3591 0.0547 0.7719 0.0819 1964 0.1136 0.3328 0.0751 0.9100 0.0867 1965 0.1520 0.3814 0.1435 1.2079 0.1387 1966 0.1371 0.6316 0.0622 1.0063 0.1587 1967 0.1217 0.5366 0.1157 0.4574 0.1748 1968 0.03035 0.4346 0.0984 0.2593 0.1824 1969 0.07905 0.4357 0.1114 0.4671 0.1790 1970 0.2463 0.3725 0.1228 0.4840 0.2459 1971 0.2347 0.3571 0.1605 0.5725 0.2538 1972 0.1681 0.4968 0.1275 0.5027 0.2498 RPAG: Commercial Ag r i cu l tu re RPMF: Manufacturing RPTR: Transport and Communications RPCN: Construct ion RPSV: Services Source: Computed using formulae (5.7) and (5.8) and data on purchase pr i ce of investment goods (Table X X X 1 1 1 ) a p a p-i t a':l c co n s um p.t i o n allowances (Table XXXVI), value added (Table XLI), the wage b i l l (der ived from Tables XLI LInanel:. XLiV)<j..a:ndmthea.tax-;rate (Table XXXVII). -127-34 p r o f i t s net of i n d i r e c t taxes. In the absence of complete data on income taxes paid by the mining industry we employ two a l t e r n a t i v e tax ra tes ; ( i ) the rate app l i cab le to the res t of the economy and ( i i ) a tax rate ca l cu la ted as 45 percent of mining p r o f i t s d iv ided by tota l p r o f i t s . (Note that the Zambian s t a t i s t i c s make no reference to dep let ion a l lowan-ces ) . Mining p r o f i t s and re la ted data are given in Table XXXIX while the d i r e c t taxes and the der ived tax rates are reported in Table XL. The pr ice index of machinery and the cap i t a l consumption allowances are found in Table XXXV and XXXVIII r e s p e c t i v e l y . F i n a l l y , the computed four a l t e r n a t i v e se rv i ce pr ices f o r machinery and equipment are reported in Table VIII from which i t can be seen that the higher the deprec ia t ion and taxat ion rates the higher the renta l pr ices of cap i t a l as we would have expected. Table VI reveals a s im i l a r s e n s i t i v i t y of the cap i t a l stocks to a l t e r n a t i v e assumptions about deprec i a t i on . Henceforth, we w i l l be using se r ie s KVC1 and KFC3 of Table VI to represent the stocks of va r i ab le and f i xed cap i t a l r e spec t i ve l y and the ser ies RPKV1 of Table VIII as the corresponding renta l p r i ce ser ies f o r va r i ab le c a p i t a l . In the absence of information on the average ' l i f e ' of machinery and s t ructures in Zambian mining the above choice i s somewhat a r b i t r a r y based on what are considered to be reasonable deprec ia t ion rates f o r machinery and s t ructures in Canadian mining (see Woodland [1972]). We w i l l return to fu r ther d i s cus -s ion of mining inputs in sect ion 5.4 fo l lowing a b r i e f review of the D i v i s i a index method and i t s app l i c a t i on in aggregating non-mining inputs -1 28-TABLE VIII COMPUTED RENTAL PRICES FOR MINING VARIABLE CAPITAL (MACHINERY AND EQUIPMENT), ZAMBIA, 1945-1970. YEAR RPKVI RPKV2 RPKV3 RPKV4 1945 0.0529 0.0471 0.0614 0.0558 1946 0.0648 0.0572 0.0778 0.0698 1947 0.0825 0.0729 0.0993 0.0889 1948 0.0931 0.0827 0.1062 0.0948 1949 0.0946 0.0839 0.1033 0.0919 1950 0.0988 0.0869 0.1084 0.0954 1951 0.1169 0.1030 0.1368 0.1204 1952 0.1341 0.1180 0.1414 0.1242 1953 0.1430 0.1264 0.1546 0.1365 1954 0.1262 0.1112 0.1532 0.1348 1955 0.1289 0.1136 0.1554 0.1368 1956 0.1487 0.1318 0.1590 0.1408 1957 0.1599 0.1412 0.1450 0.1282 1958 0.1556 0.1395 0.1650 0.1480 1959 0.1457 0.1312 0.1885 0.1698 1960 0.1639 0.1479 0.2015 0.1818 1961 0.1841 0.1659 0.2097 0.1890 1962 0.1829 0.1648 0.2078 0.1872 1963 0.1878 0.1688 0.2190 0.197(11 1964 0.1926 0.1735 0.2093 0.1886 1965 0.1678 0.151(11 0.1720 0.1549 1966 0.2378 0.216il) 0.2220 0.2015 1967 0.2493 0.2278 0.2487 0.2272 1968 0.2771 0.2552 0.2498 0.2287 1969 0.2812 0.2613 0.2790 0.2591 1970 0.3674 0.3445 0.2809 0.2590 RPKV1 Computed with 6 = .(110 and TAXR (See Table XL) ' ,1• RPKV2 Computed with •&= .08 and TAXR RPKV3 Computed with .6= .10 and TAXR2 (See Table XL) RPKV4 Computed with s .08 and TAXR2 6 One-period constant deprec iat ion rate TAXR? Income tax rate i d e n t i c a l to that o f the non-mining sector TAXR2 45 percent of mining p r o f i t s [net of i n d i r e c t taxes) d iv ided by to ta l p r o f i t s . Source: Computed using formula (5.7) and data given in Tables -XXX.V, J(L c-and :XXXV.LT.I). -1 29-and outputs ( sect ion 5.3). 5.3 Aggregation of Non-mining Inputs and Outputs: The D i y i s i a Index In the absence of aggregate non-mining se r ie s on inputs and outputs f o r Zambia we employ the D i v i s i a index to aggregate output, c a p i t a l , and 35 labor over the f i v e non-mining subsectors. The D i v i s i a [1926] index has been extens ive ly discussed and appl ied to the measurement of to ta l f a c to r p roduc t i v i t y by Solow [1957], R ichter [1966], Jorgenson and G r i l i c h e s [1967]. T h e o r e t i c a l l y the most appropr iate index fo r aggregation is the continuous time D i v i s i a index but fo r emp i r i -cal app l i ca t i ons some d i s c re te approximation must be used, as the data come in d i s c r e t e form. In what fo l lows we def ine the continuous time D i v i s i a index and the Tornqvis t d i s c re te approximation to th i s index. For a more de ta i l ed expos i t ion see Wold and Jareen [1953] and Diewert [1975] [ 1977] . Considering i n f i n i t e s i m a l quant i ty changes ( in inputs or outputs) from Q(0) = [ Q 1 ( 0 ) , . . . , Q n ( 0 ) ] in time 0 to Q(l) = [ Q n ( 1 ) , . . . , Q n ( l ) ] at time 1 and the corresponding p r i ce changes from P(0) = [ P-|-(0),... ,P (0)] to P ( l ) = [ P ^ ( l ) » . . . P (1)] we may def ine the growth ra te of the continuous time D i v i s i a quant i ty index by the fo l lowing d i f f e r e n t i a l equation: nm n M * ) ?[lr= ^ s i ( t ) t j j r t r 1 = 1 n <5-10) n where Q, = dQ/dt and S.(t) = P . ( t )Q( t ) / z P. ( t )Q.(t ) i s a weight given by the r e l a t i v e share of the value of the i t h input (or output) in the value of tota l input (or output) . -130-Given continuous data on output and pr ices we can f i n d . t h e d e f i n i t e integra l of (5.10) to obtain the D i v i s i a quant i ty index, Q ( t ) , rather than the growth rate of the index given by (5.10). Integrating (5.10) with respect to time from the i n i t i a l period 0 to the terminal period T we obta in: Q(T) = Q(0) exp {fl I S . ( t ) [Q . ( t ) /Q ( t ) ]d t } (5.11) u t=l 1 1 The continuous time D i v i s i a quant i ty index as def ined by (5.11) is the t h e o r e t i c a l l y most appropr iate index f o r aggregation of inputs and outputs. Economic data, however, come in d i s c re te form and thus, a d i s c re te approx i -mation to the continuous D i v i s i a quant i ty index must be used. There are many such d i s c r e t e approximations such as the Laspeyres, Paasche, F isher and Tornqvis t indexes, but un t i l r e cen t l y , we had no way of d i s c r im ina t ing among them. Diewert [1975] using the economic theory of exact index numbers has shown that the F i sher and Tornqvis t indexes are the t h e o r e t i c a l l y most appropr iate d i s c re te approximations to the continuous time D i v i s i a quant ity index. In what fo l lows we employ Tornqv is t [1936] quant i ty and p r i ce indexes. The Tornqvis t quant i ty index i s derived as a d i s c re te approximation to (5.11), as fo l lows: f o r T = 1 and S^O) = S ^ l ) the logarithm of the D i v i s i a index i s obtained as: n 1 0 £ n [ Q ( l ) / Q ( 0 ) ] = E S . A [Q./Q ] ; i=l i n l i but s ince in general S..(0) f S ^ l ) Tornqv is t [1936] suggested the fo l lowing -131-d i s c r e t e approximation to the continuous D i v i s i a quant ity index: inO] - £nQ° = 1 _E (S1. + sj) UnQ] - inQ°.) with S ° % P ° Q°/E P ° Q ° 1 1 V •_•] 1 H l 1 1 1 n 1 1 and S = P. Q / E p. Q 1 1 1 i=i 1 1 or , more genera l l y , n t mQ1 - £ n Q t _ 1 = E W. UnQ* - JinQ^"1) (5.12) i=l 1 1 1 with w h l f s J + S ^ 1 ) t t t n t t and S. = P. Q 7 E P.Q: I I i -j = i i i where Q-^ : the quant i ty of output ( input) in the i th sector in period t , P*: the corresponding p r i ce of output ( input) in the i t h sector W :^ ar i thmet ic weights; and S*: the r e l a t i v e share of the i t h s ec to r ' s output ( input) in period t The Tornqvis t p r i ce index can be def ined by a formula analogous to (5.12) by rep lac ing Q by P: znP1 - £ n P t _ 1 = E W* UnP^ - J>nP*_1) (5.13) i=l 1 1 1 Thus the Tornqvis t d i s c r e t e approximation to the continuous D i v i s i a index uses ar i thmethic weights to weight the changes in the logarithms of the sectora l p r i ce and quant i ty indexes in order to compute the changes in the -132-logarithms of the d i s c re te D i v i s i a p r i ce and quant ity indexes. These two aggregate indexes can be derived independently using formulae (5.13) and (5.12) r e s p e c t i v e l y . There is a l so the option that only one of them i s der ived e x p l i c i t l y through the above process while the other i s der ived i m p l i c i t l y using F i s h e r ' s [1922] weak fac tor reversa l t e s t . The l a t t e r states that the product of the pr i ce and quant i ty indexes should equal the r a t i o o f tota l expenditures in the two per iods. In construct ing aggregate quant i ty ser ies to be used in the est imation of the non-mining production funct ion we choose to der ive the quant ity index e x p l i c i t l y using formula (5.12) and adopting the convention that: The aggregate ind ices Q and P are obtained a f t e r normal izat ion fo r some base period (1967 in the present s tudy). To cons t ruct an aggregate output se r ie s f o r the e n t i r e non-mining sector f o r the period 1954-72 using the above procedure, we need f i gures on Gross Domestic Product (G.D.P.) at f ac to r cost fo r A g r i c u l t u r e , Manufacturing, Transport , Construct ion, and Services and the corresponding p r i c e d e f l a t o r s . These are obtained from McPherson [1976a: Tables G.1..1 and D . l . l ] and reproduced in Tables XLI and XL 11. For aggregating cap i t a l over the f i v e non-mining sectors we use the sectora l cap i t a l stock f i gures n t t (5.14) Then the p r i ce index i s der ived i m p l i c i t l y as: (5.15) -133-(only f o r the years 1954-72) constructed in sect ion 5.1 and reported in Table V and the corresponding renta l p r i ce constructed in sect ion 5.2 and reported in Table VII. F i n a l l y , fo r the construct ion of a labor force aggregate we use the ten employment se r ie s fo r A f r i cans and Non-Afr icans f o r the f i v e sectors constructed by McPherson [1976: Table 11.5] and the corresponding average annual earnings a l so reported in McPherson [ 1976: Table E . l . l ] and reproduced in Table XL 111 and XLIV. The aggregate, D i v i s i a quant i ty ind ices fo r output, c ap i t a l and labor and t h e i r respect ive i m p l i c i t p r i ce ind ices are reported in Table IX. 5.4 Mining Inputs and Outputs: Aggregation of Imported Var iab le Inputs Est imation of the un i t v a r i ab l e cost funct ion and the corresponding cost share equations requires time ser ies on ( i ) the pr i ce of va r i ab le inputs and the quant i t i e s of f i xed inputs as independent var iab les and ( i i ) the un i t va r i ab le cost and cost shares as dependent var iab les which, in tu rn , requ i re time se r ie s on pr i ces and quant i t i e s of v a r i ab le inputs and quant i ty of output. Up to th i s point we have constructed ser ies on va r i ab le and f i xed cap i t a l ( sect ion 5.1) and on rental pr ices of va r i ab le cap i t a l ( sect ion 5.2). There are two. more va r i ab le inputs , domestic and expatr ia te l abor , and one more f i x e d input , the copper resource stock. This sect ion deals f i r s t with output, then with va r i ab le inputs, and l a s t with f i xed inputs . Zambia produces and exports a number of minerals : copper, c o b a l t , z i n c , lead and c o a l . Mining output could be obtained by aggregating the ind iv idua l quant i ty ser ies using the i r corresponding p r i ce ind ices and employing the D i v i s i a index discussed in the previous 37 sec t i on . Instead, we use the annual copper production ser ies to represent -134-TABLE IX DIVISIA QUANTITY AND IMPLICIT PRICE INDICES FOR NON-MINING OUTPUT, CAPITAL AND LABOR, ZAMBIA, 1954-1972. (1967 Base Year). YEAR Y l . K N pv PK P W 1954 0. 2805 0 .2178 0. 7320 307. 0150 141. 5756 79. 9093 1955 0. 3297 0 .2772 0. 7661 321. 1750 161. 1506 85. 1011 1956 0. 3918 0 .3377 0. 8453 328. 3730 164. 1693 94. 2573 1957 0. 4347 0 .4014 0. 9051 329. 5890 155. 5155 99. 8190 1958 0. 4460 0 .4432 0. 9279 326. 1020 118. 4356 104. 4120 1959 0. 4681 0 .4931 0. 8877 333. 3630 122. 5845 n o . 7030 1960 0. 4761 0 .5238 0. 8779 343. 2700 129. 1167 114. 6620 1961 0. 4889 0 .5473 0. 8558 346. 5420 134. 8605 120. 3110 1962 0. 4945 0 .5626 0. 8468 351. 7310 130. 8440 127. 9640 1963 0. 5030 0 .5747 0. 8442 361. 0500 134. 0774 131. 5290 1964 0. 5489 0 .6049 0. 8373 367. 3090 142. 5775 145. 8690 1965 0. 7600 0 .7215 0. 9261 390. 2880 214. 2445 162. 3710 1966 0. 8243 0 .8472 0. 9700 432. 4010 227. 3206 180. 7610 1967 1. 0000 1 .0000 1. 0000 447. 0000 227. 2978 230. 6300 1968 1. 0382 1 .1705 1. 0302 473. 6170 209. 8673 253. 4260 1969 1. 0540 1 .2765 0. 9985 498. 6600 220. 1907 268. 6780 1970 1. 2462 1 .4255 1. 0324 513. 7030 278. 1924 281. 7190 1971 1. 3238 1 .5384 1. 0700 541. 8940 291. 3630 322. 7390 1972 1. 3738 1 .6473 1. 1128 586. 8440 290. 4370 318. 8690 Y-|:: Aggregate non-mining output K: Aggregate non-mowing cap i ta l stock N: Aggregate non-mining labor Py.: Imp l i c i t p r i ce index of output Pj|: Imp l i c i t p r i ce index of cap i t a l Pft: Imp l i c i t p r i ce index of labor Source: Constructed using the D i v i s i a Index Method discussed in Sect ion 5.3 and sectora l quant i ty and pr i ce ser ies on output (Tables XL'Va'nd "WtM-} labor (Table XL III nrindLXli IV*). ntfnd £ca" p<i ta 1 (Tables V and VI I). -135-mining output, on the grounds that copper accounts fo r more than 95 percent 38 of the to ta l value of annual mineral product ion. In general the present study treats the copper industry as equiva lent to the mining sector . Copper production is a l so treated as equiva lent to copper exports, in the absence of s i g n i f i c a n t domestic copper consumption or s t o c k - p i l l i n g within 39 Zambia. Output ser ies are given in a number of sources, inc lud ing Metal S t a t i s t i c s , the Mi ni ng Yearbook of Zambia and Coleman [1971]. The ser ies on output quant i t i e s and pr ices fo r the period 1925-72 are reported in Table X. While only the quant i ty ser ies fo r 1945-70 are used in obta in ing the un i t cost f i gures the en t i r e ser ies i s employed in construc-t ing cumulative production f i gures in the next sec t i on . Note that a l l quant i ty f i gures have been converted to metric tons and a l l p r i ce f i gures to kwachas fo r c o n s i s t e n c y . 4 0 The mining industry employs the fo l lowing va r i ab le inputs: ( i ) va r i ab le cap i t a l or machinery and equipment, ( i i ) energy and mater ia l s and ( i i i ) l abor . In the case of Zambia there was no ava i l ab l e se r ie s on energy and mater ia l s except for few scattered observat ions. Most const ruct ion mater i a l s , however, are included in f i xed cap i t a l as part of bu i ld ings and mine development whi le , at l e a s t , some important mater ia l s 41 are part of the va r i ab le c a p i t a l . No s i g n i f i c a n t quant i t ie s of other mater ia l s are used in copper mining except, of course, f o r the copper ore in the ground, which, however, i s treated here as a f i xed input (see sect ion 5.5 below). The s i t ua t i on with regard to energy i s even less s a t i s f a c t o r y . While we know that e l e c t r i c i t y makes up 70 percent of to ta l 42 use of energy and that coal and fuel account f o r the res t no comprehensive -136-TABLE X COPPER PRODUCTION AND COPPER PRICES, ZAMBIA, 1924-1972. (Output: in metric tons .P r i ces : in Kwachas per metric ton) . YEAR QCU PCU YEAR QCU PCU 1924 0.10 118.08 1950 280.86 352.30 1925 0.10 127.33 1951 314.09 434.30 1926 0.71 125.17 1952 317.36 510.70 1927 3.05 117.10 1953 368.39 504.60 1928 6.09 132.05 1954 389.00 454.00 1929 5.08 164.33 1955 358.60 637.00 1930 6.09 117.69 1956 404.10 609.00 1931 9.14 73.60 1957 435.70 388.00 1932 69.09 50.38 1958 400.10 311.00 1933 105.66 63.76 1959 543.30 412.00 1934 139.80 76.36 1960 576.40 430.00 1935 146.30 79.11 1961 574.70 404.00 1936 144.27 85.80 1962 562.30 410.00 1937 150.37 119.46 1963 588.10 410.00 1938 216.41 90.7Z, 1964 632.20 434.00 1939 215.39 99.580. 1965 695.70 502.00 1940 266.59 125.17 1966 623.40 769.00 1941 231.90 122.80 1967 663.00 722.00 1942 250.54 122.02 1968 684.90 803.00 1943 254.97 122.02 1969 719.50 992.00 1944 224.36 122.02 1970 684.10 996.00 1945 197.19 122.02 1971 651.40 709.00 1946 185.21 151.93 1972 717.10 690.00 1947 195.58 257.02 1948 217.05 263.71 1949 263.23 261.74 QCU: Copper output PCU: Average export p r i ce of Zambian copper Sources: Output: 1924, 1938-53 Coleman [1971] 1925-38 Barber [1961] 1953- 72 McPherson [1976b 116, D.2.13] 1973-75 Mindeco Mining Yearbook of Zambia P r i ce s : 1924-53 Coleman [1971] 1954- 72 McPherson [1976b: C . l . l ] -137-data on the use of these inputs have.been loca ted. FrQJD scattered i n f o r -mation, however, i t appears that before the energy c r i s i s of the ear l y 1970's, energy expenditure was not a major part ( c e r t a i n l y le s s than 43 10 percent) of the to ta l cost of product ion. Thus, we ignore the energy input and estimate a va r i ab le cost f unc t i on , which i s net of energy 44 cos t . The Zambian labor s t a t i s t i c s d i s t i n gu i sh between loca l (Afr ican), and expatr iate,(European) l abor . I n i t i a l l y , t h i s was analogous to the con-ventional d i s t i n c t i o n between s k i l l e d and unsk i l l ed labor but increas ing ly more A f r i cans are rep lac ing Europeans in semi - sk i l l ed and s k i l l e d pos i t i ons . The time ser ies f o r the two groups of labor and the respect ive average annual earnings are reported in Table XI. The expatr iates are mainly technic ians operat ing the complicated imported machinery and managers and engineers superv is ing the mining operat ions. They rece ive wages and s a l a r i e s several times those paid to loca l labor (see Table XI) and they send abroad a substant ia l port ion of t h e i r annual earnings. In f a c t they may be thought of as imported inputs , complementary to machinery and equipment and be combined with the l a t t e r into one imported va r i ab le input. In the words of Mikesel l [1970] "the technica l and managerial 45 serv ices are in e f f e c t j o i n t costs with the cap i ta l investment". Thus, to maintain adequate degrees of freedom for econometric est imation of a 46 f l e x i b l e funct iona l form f o r v a r i a b l e co s t , we aggregate va r i ab l e cap i t a l and expat r ia te labor ( i . e . machinery and technica l personnel) into one 47 imported va r i ab l e input, using the D i y i s i a index discussed in the 48 previous sec t i on . The aggregate quant i ty , M, and pr i ce index, P M , of the -138-TABLE XI MINING EMPLOYMENT OF AFRICANS AND EUROPEANS AND THEIR RESPECTIVE AVERAGE ANNUAL EARNINGS, ZAMBIA, 1945-1970. (Employment in '000 of Persons; earnings in '000 of Kwachas). YEAR WACU WECU LACU LECU 1945 66.00 1472.00 33.50 3.30 1946 70.00 1584.00 31.50 3.20 1947 76.00 1847.00 34.00 3.80 1948 94.00 2003.00 36.50 4.20 1949. 104.00 2112.00 38.25 4.50 1950 122.00 2136.00 40.000 4.80 1951 156.00 2550.00 42.00 5.50 1952 172.00 3000.00 44.00 5.80 1953 248.00 3564.00 46.00 5.90 1954 264.00 4240.00 44.10 6.80 1955 312.00 4866.00 42.50 7.20 1956 368.00 4858.00 46.20 7.70 1957 410.00 4348.00 47.40 8.10 1958 462.00 4392.00 39.90 7.40 1959 . 530.00 4894.00 41.20 7.80 1960 570.00 5188.00 42.70 8.00 1961 578.00 5178.00 42.10 8.10 1962 592.00 5126.00 41.10 8.30 1963 596.00 5128.00 40.80 8.20 1964 732.00 5150.00 42.40 8.30 1965 826.00 5378.00 44.80 7.50 1966 934.00 6598.00 47.60 7.20 1967 1322.00 7608.00 48.30 6.40 1968 1248.00 7604.00 48.60 6.10 1969 1412.00 8174.00 50.30 5.60 1970 1543.00 7229.00 52.10 5.50 LACU: Mining employment ©if? A f r i cans LECU: Mining employment of European expatr iates WACU: Average annual earnings of A f r icans WECU: Average annual earnings of Europeans Sources : 1954-70: McPherson [1976b: Tables II.5, E . l . l ] 1954-53: Table XLV of Appendix C. -139-imported va r i ab le input are reported in Table XII and combined with the quant i ty , L, and p r i c e , P^, o f the domestic va r i ab le input (Afr ican labor) to construct the to ta l va r i ab le co s t , C, s e r i e s . Then, the un i t va r i ab le cos t , c , i s obtained by d i v i d i ng to ta l cost by output, Q: c = C/Q = (P M - M + P L • L)/0 (5.16) 49 The tota l and un i t va r i ab le cost and the un i t quant i t ie s of the two va r i ab le inputs are recorded in Table XII. F i n a l l y , two f i xed inputs enter the mining production process act ing as s h i f t parameters on the cost func t ion : ( i ) the f i xed physical c a p i t a l ; that i s , bu i ld ing and engineering s t ruc tu re s ; and ( i i ) the f i xed natural c a p i t a l ; that i s , the copper resource stock remaining in the ground at each point in time. For the former, stock f i gures have been constructed in sect ion 5.1 and reported in Table VI. For the l a t t e r , stock ser ies are constructed in sect ion 5.5 below. 5.5 Est imation of the Copper Resource Stock The minera l ized rock (ore) and i t s copper content wi th in Zambian t e r r i t o r y i s a non-renewable f i n i t e resource. A substant ia l part of th i s resource ( e spec i a l l y in terms of copper content) has a lready been removed by 40 years of l a rge sca le mining operat ions. We, then, need to know how much of Zambia's i n i t i a l copper endowment remains; that i s , the s i ze of the present copper resource stock. The s i ze of th i s stock a f f e c t s i t s user cost and consequently i t s rate of dep le t i on . Furthermore, the s i ze of the resource stock a t each point in time a f fec t s both ex t rac t ion and treatment cost : as the stock i s ' s h r i n k i n g ' , the va r i ab le cost of produc-t ion r i s e s because of both d iminish ing a c c e s s i b i l i t y of the ore and recource -140-TABLE X I I D I V I S I A P R I C E I N D E X O F I M P O R T E D V A R I A B L E I N P U T A N D C O R R E S P O N D I N G I M P L I C I T Q U A N T I T Y I N D E X , T O T A L V A R I A B L E C O S T , U N I T C O S T A N D U N I T Q U A N T I T I E S O F V A R I A B L E I N P U T S I N M I N I N G , Z A M B I A , 1945-1970. YEAR PKVL1 KVLE1 COST c m I 1945 0.2057 30.9380 8.57585 0. 0434 0. 1567 0. 1697 1946 0.2288 30.1854 9.1101' 0. 0490 0. 1628 0. 1699 1947 0.2730 33.3718 11.6197 0. 0592 0. 1704 0. 1686 1948 0.2994 38.9733 15.0983 0. 0694 0. 1793 0. 1680 1949 0.3124 43.0363 17.4159 0. 0660 0. 1633 0. 1450 1950 0.3191 48.3143 20.2988 0. 0721 0. 1718 0. 1423 1951 0.3798 56.3050 27.9389 0. 0877 0. 1790 0. 1336 1952 0.4428 61.1178 34.2853 0. 1078 0. 1924 0. 1322 1953 0.5048 67.2486 45.3560 0. 1229 0. 1824 0. 1247 1954 0.5401 79.7478 54.7108 0. 1404 0. 2048 0. 1133 1955 0.5967 86.3558 64.7875 0. 1803 0. 2406 0. 1184 1956 0.6260 93.8509 75.7517 0. 1871 0. 2320 0. 1142 1957 0.6021 101.5600 80.5843 0. 1846 0. 2329 0. 1087 1958 0.5984 98.1077 77.1424 0. 1924 0. 2450 0. 0996 1959 0.6199 101.8270 84.9574 0. 1573 0. 1887 0. 0763 1960 0.6730 104.6550 94.7708 0. 1641 0. 1814 0. 0740 1961 0.7067 108.1890 100.7930 0. 1750 0. 1881 0. 0732 1962 0.7009 111.1270 102.2150 0. 1814 0. 1974 0. 0730 1963 0.7094 110.2020 102.4960 0. 1739 0. 1872 0. 0693 1964 0.7194 110.3440 110.4220 0. 1743 0. 1743 0. 0670 1965 0.6922 104.4460 109.3030 0. 1592 0. 1522 0. 0653 1966 0.9077 100.8580 136.0040 0. 2177 0. 1616 0. 0763 1967 1.0000 93.3709 157.2230 0. 2367 0. 1407 0. 0728 1968 1.0525 87.7622 153.0260 0. 2230 0. 1280 0. 0709 1969 1.0991 86.1599 165.7190 0. 2299 0. 1196 0. 0698 1970 1.2114 84.6304 182.9080 0. 2668 0. 1236 0. 0760 P|V|=PKVL1: D i v i s i a p r i ce index of imported var i ab le input M (aggregate 5 of renta l p r i ce of var iab le cap i t a l and European wages) M E K V L E I : Corresponding i m p l i c i t quant ity index 'csC/QCU: Unit va r i ab le cost (cost d iv ided by output QCU given in Table: X ) mEM/QCU: Unitequant i ty of KVLE1; £ H L / Q C U : Unit quant ity of A f r i can Labor (see Table X I ) . Sources: Computed from data given in Table V I I I , X and X I . -141-to lower grades. The copper resource stock (or simply copper resource) and the ore resource stock (or simply ore resource) :are to be d i s t ingu i shed from the copper and ore reserves maintained, updated, and published by the mining companies. Reserve f i gures correspond ne i ther to the to ta l volume of ore that can be extracted over the l i f e of a mine nor to the tota l amount of copper that can be produced. If they d i d , the Zambian Copperbelt would have been exhausted years ago. In r e a l i t y , Zambian reserves today, a f t e r 40 years of e x t r a c t i o n , are double t h e i r 1930's s i z e . Company estimates inc lude , " f i r s t , only that mater ia l whose presence has been demonstrated with cons iderable assurance, and second only material s u f f i c i e n t l y r i c h 51 to support p r o f i t a b l e operat ion" under the economic condi t ions of 52 the time. '. Since economic condit ions change over time, ore reserves are a most e lu s i ve concept; small va r i a t i ons in the p r i ce / co s t r e l a t i o n may cause reserves to be wr i t ten up or down; the higher the copper pr ices and the lower the cos t , the larger the copper reserves and v i ce versa. What we need i s an estimate of the copper resource stock which i s independent of the economics o f the time and takes into account not only proven material but the ul t imate resources of a given geologic e n t i t y . Fol lowing Zwartendyk [1972] we def ine the 'copper resource ' to include known and unknown stocks of copper that may or may not be economical ly p r o f i t a b l e , but seem l i k e l y to become in the future under c e r t a i n s p e c i f i e d cond i t i ons . The copper resource includes both present reserves and materia l which w i l l be sh i f t ed into the reserve category in the forseeable fu ture through d i scovery , technica l change and higher p r i ce s . Tonnage and -142-grade of unknown deposits are based mainly on knowledge of the charac-t e r i s t i c s of known deposits with in the same mineral area. The question then becomes, how can we use ex i s t i ng data on ( p a r t i a l l y ) known deposits to i n f e r the tonnage and grade of mater ia l remaining in these depos i t s , or other t o t a l l y unknown deposits wi th in the same geological area? Although ore reserves alone t e l l us very l i t t l e about mineral resources they are not t o t a l l y meaningless. A long h i s to ry of reserve estimates and corresponding grades when combined with the record of past production can 53 in f a c t determine the 'u l t imate mineral resource 1 of a given geolog ica l e n t i t y . The 'u l t imate mineral resource 1 i s defined as the tota l amount of the metal content of the cont inenta l c rus t concentrated in ore deposits (not in s o l i d so lu t ion in s i l i c a t e minerals) and therefore amenable to 54 t r a d i t i o n a l mining, benef i ca t ion and smelt ing. Thus, the u l t imate copper 55 resource of Zambia (or i t s i n i t i a l copper endowment), i s the tota l copper content of i t s t e r r i t o r y ( inc lud ing past production) found in mineral concentrat ions higher than that of the common rock. Once the u l t imate copper resource has been determined i t s unexploited part at each po int time (what we termed copper resource stock) can be def ined by subtract ing the cumulative production to that date from the t o t a l . Thus, the remainder of th i s sect ion w i l l be devoted to the est imation of the ul t imate copper resource of Zambia and i t s d e r i v a t i v e s e r i e s , the copper resource stock f o r the years 1945-1975, and the average grade of cumulative production for the same per iod . In estimating the i n i t i a l copper endowment of Zambia we extend Lasky 's [1950] approach to incorporate Sk inner ' s [1976] [1977] con t r i bu -56 t ion of using ' the energy b a r r i e r ' as the resource c u t - o f f grade. -143-;Before introducing Lasky 's approach we, f i r s t c l a r i f y the concept of the 'energy b a r r i e r ' and i t s ro le in d i s t i ngu i sh ing ore from common rock, as developed by Skinner. Skinner [T976] [1977] d i s t ingu i shes metals according to the i r "geochemical abundance", def ined as the percentage weight of each metal in the ear th ' s cont inenta l c ru s t . Geochemically abundant metals are 57 termed those present in amounts above 0.1 percent, such as aluminum, i r o n , and magnesium and geochemically scarce are those present in amounts below 0.1 percent such as copper (0.0058 percent ) , lead (0.0010 percent) and t i n (0.00015 percent)'. Geochemical l y abundant metals are found in ore deposits that grade slowly into common rock from which " i t i s always poss ib le to produce a mineral concentrate in which the des ired abundant element i s a major c o n s t i t u e n t . . . The smelting pract i ces C O ava i l ab l e today w i l l a lso work in the fu tu re " . The geochemically scarce metals on the other hand, are found in two d i s t i n c t forms: ( i ) as separate minerals in the rare l o c a l i z e d ore deposits formed under very unusual circumstances ( i i ) as chemica l ly dispersed atoms enclosed with in the s t ructures of s i l i c a t e minerals in common rock. The 59 f i r s t form i s amenable to concentrat ion p r i o r to smelt ing, as i s genera l ly the case with abundant metals, but un l ike the l a t t e r , ore deposits of scarce metals do not grade slowly in to common rock. There appears to e x i s t a sudden d i s con t i nu i t y between the two modes of occurence marked 60 by the enormous energy requirements of breaking down chemical ly the atomic s t ructure of the en t i re mineral to separate the atoms of a scarce metal from a l l other atoms. Concentration of chemical ly dispersed atoms of minerals p r i o r to smelting i s not pos s ib le . -144-Returning to copper, the most widely used scarce metal , we can break down the problem of measuring the ult imate copper resource of a given area into two separate quest ions: ( i ) a t what grade is the energy ba r r i e r reached in the case of copper? and ( i i ) how much of the given areas copper content i s concentrated in deposits of that grade or higher? A recent report of the Committee of Mineral Resources and the Environment of the U.S.A. National Academy of Sc iences, COMRATE [1975], estimated that the energy ba r r i e r fo r copper i s reached at a grade of 0.1 percent and that "no more that 0.01 percent of the to ta l copper in the cont inenta l c ru s t w i l l be found concentrated in ore bodies with grades of 0.1 percent copper or 61 more" . Thus, in answering question ( i i ) above we extend Lasky 's approach [1950] to incorporate COMPATE-Skinner's energy ba r r i e r as the resource c u t - o f f grade. Lasky [1950] suggested that the u l t imate mineral resource of a given geologic e n t i t y can be der ived from the grade-tonnage pat tern , revealed in the sample of past production and ore reserve est imates, in the same way that the number of holes from a shot-gun discharge contained in the outer c i r c l e of a large target can be ca lcu la ted from the pattern of holes in the ins ide c i r c l e of the target . Lasky, a f t e r studying a va r i e ty of minera l s , and in p a r t i c u l a r the porphyry copper deposits in the U.S., came to the conc lus ion tha t , . . . i t may be stated as a general p r i n c i p l e that in many mineral deposits in which there is graduation from r e l a t i v e l y r i c h to r e l a t i v e l y lean material the tonnage increases at a constant geometric ra te as the grade decreases. Or as the tonnage increases, the mineral content of the cumulative tonnage decreases a t a constant dec l i n ing r a te . -145-On the e x p l i c i t assumption that best grades are mined f i r s t Lasky s p e c i f i e s the fo l lowing r e l a t i o n s h i p between cumulative ore tonnage, D^, and i t s copper content, A^: -bD A t = A o - A o . e (5.17) where A Q and b are constants to be estimated f o r each geologic e n t i t y . Equation (5.17) i s obtained as a spec ia l case of the general ized exponential f unc t i on : y = c + a e K X (5.18) by r e s t r i c t i n g a < 0, k < 0, c > 0, c = |a| and subs t i tu t ing A t f o r cn y, D^ for x, -b fo r k, A Q f o r c and -A Q f o r a. The cumulative ore tonnage at time t , D^, i s def ined as the sum of past ore production cumulated at time t and ore reserves estimates at time t . The copper content of D^ or cumulative copper tonnage, A^, i s def ined as the sum of past copper production cumulated to time t and the copper content of the ore reserves at time t . The l a t t e r i s obtained by simply mu l t ip ly ing ore reserves by the corresponding average grade adjusted f o r i r r e d u c i b l e technolog ica l losses in recovery. The use of cumulative production plus reserves f i gures i s based on the presumption that: When the material estimated as reserves at any p a r t i c u l a r time is added and averaged with the ore produced to that time, the re su l t i ng f i gu re can be said to approximate the tota l amount of mater ia l i n the depos i t at that average grade. 5 Equation (5.17) may be depicted g raph i ca l l y as the 'curve of diminishing increments' of f i gu re 1 i nd i ca t ing that: ( i ) the cumulative copper -146-FIGURE 1. RELATION BETWEEN CUMULATIVE ORE TONNAGE AND CUMULATIVE COPPER TONNAGE: THE "CURVE OF DIMINISHING INCREMENTS" Cumulative ore tonnage up to time t t Notat ion: A q : u l t imate copper resource D : u l t imate ore resource -147-tonnage i s a monotonical ly increas ing funct ion of the cumulative ore tonnage; and ( i i ) as the cumulative ore tonnage increases i t s marginal cont r ibut ion to the cumulative copper tonnage diminishes. The constant A i s the v e r t i c a l d i s tance of the curve from the hor izonta l axis at the point where the slope of the curve becomes zero. As the slope of the curve approaches zero l e s s and less copper content i s added to A^ by each add i t iona l ton of ore. Eventual ly a point i s reached where the curve a t ta in s a maximum at which a l l mineral has been inc luded, and from there on only barren materia l can be added. At th i s point A. = A . In th i s J ^ t o sense A Q i s the ul t imate copper resource (or equ iva len t l y , the i n i t i a l copper endowment) of the geologic en t i t y under cons idera t ion . This can be seen by d i f f e r e n t i a t i n g (5.17) with respect to D and se t t ing dA/dD equal to zero: Since b > 0, neces sa r i l y A - A = 0 and A. = A . J o t t o Given a long h i s to ry of past production and reserve est imates, time ser ies on A^ . and can be constructed, and given appropriate s tochast ic s p e c i f i c a t i o n for equation (5.17), empir ical estimates of b and A„ can be obtained (see below for d e t a i l s on data const ruct ion and est imation) Once the u l t imate copper resource, A , i s estimated the u l t imate ore resource, D , can be e a s i l y obtained from (5.17) by so lv ing fo r D. o A J = 0 (5.19) o D t = - £ n (1 - A t /A Q ) /b (5.20) - 1 4 8 -As A tends to A q , tends to D q . T h e o r e t i c a l l y , however, the curve ( in f i gu re 1 ) approaches a maximum only asymptot ica l l y . Since a zero cut o f f grade i s i m p l i c i t l y assumed, D w i l l inc lude substant ia l quant i t ie s 6 7 of common rock cont r ibut ing very l i t t l e to the u l t imate copper resource. For p rac t i ca l purposes we want to know the tonnage of ore cont r ibut ing the most metal and exclude a l l materia l not amenable to t r a d i t i o n a l techniques of concentrat ion and smelt ing. In order to d i s t i n gu i sh ore from common rock we introduce the energy b a r r i e r of 0 . 1 percent of copper content suggested by COMRATE [ 1 9 7 5 ] and Skinner [ 1 9 7 6 ] [ 1 9 7 7 ] as the resource c u t - o f f grade fo r copper. By se t t ing dR/dS in equation ( 5 . 1 9 ) equal to 0 . 0 0 1 and so lv ing f o r R ,^ we obta in : A t = A q - b " 1 ( 0 . 0 0 1 ) = A Q ( 5 . 2 1 ) which gives a truncated A Q as the upper l i m i t on A^. The corresponding truncated D Q can be e a s i l y obtained from equation ( 5 . 2 0 ) . Note that the in t roduct ion of the 0 . 1 percent resource c u t - o f f grade is .expected to leave the copper stock A Q l a r ge l y unaf fected, while cut t ing down subr s t a n t i a l l y the ore stock D Q due to the very low copper content of the 6 8 common rock which is now excluded. Thus, fo r p rac t i ca l app l i ca t ions e i the r A Q or A q can be used as the u l t imate copper resource without not iceab le d i f f e r e n c e in re su l t s but only D q can be considered as the ut l imate ore resource of a given mineral area. Having obtained the i n i t i a l ore endowment, D q of a given geologic e n t i t y and i t s copper content, A Q , we can e a s i l y obtain the corresponding amounts l e f t in the ground at each point in time by simply subtract ing -149-the copper and ore produced p r i o r to - that time: t S. = D = E H. (5.22) * 0 i=0 1 t R t = A - E Q (5.23) r 0 i=0 1 where S i s the ore resource stock ( in the ground) at time t , and R i t s copper content, e a r l i e r def ined as the copper resource stock. H. and Q. t 1 1 re spec t i ve l y are the annual ore and copper output while E H. and i=0 1 t E Q. are the corresponding cumulative outputs. The r a t i o of R. to S i=0 1 1 z gives the corresponding average grade of the resource, : G t = R t / S t ( 5 ' 2 4 ) t t S i m i l a r l y the r a t i o of E Q. to E H. y i e l d s the average grade of i=0 1 i=0 1 cumulative product ion, G^ . , „ t t G? = E Q./ E H. (5.25) * 1=0 1 1=0 1 The use of the extended Lasky approach in est imating the ult imate (and current) copper resource of Zambia requ i res : ( i ) that Zambia's copper resource i s concentrated in a s ing le geologic e n t i t y ; ( i i ) that the s t ructure of the ore i s such that can be separated into chunks or blocks so that best grades can be (and in f a c t are) mined f i r s t and ( i i i ) that there are long production records and h i s to ry of reserves a v a i l a b l e . These requirements w i l l be discussed in the above order. V i r t u a l l y a l l s i g n i f i c a n t Zambian copper mines are found within an -150-69 area of 80 by 30 miles - the Copperbelt proper. The consistency of the s t r a t i g raph i c horizon and the uni formity of m inera l i za t i on throughout the Copperbelt led geolog is ts to regard i t as a s ing le geologic e n t i t y . ^ The orebodies of the Copperbelt are c l a s s i c examples of the bedded type of deposit with zonal arrangement of the ores: "The r e l a t i v e pos i t i on of the ore horizon is constant throughout in that i t occurs in sand-stone (or arkose) , sha les , quar tz i te or impure dolomite of the Lower R o a n . " ^ Reeve [1963] a l so reports that Ga r l i c k , consu l t ing Geologist of the RST group of companies operating on the Copperbelt suggested the fo l lowing pattern as one of the features of the d i s t r i b u t i o n of Zambian sy lphide ores: "As a r e s u l t of th i s zonal arrangement of the su lph ides . . . the r i c h e r copper tends to be associated with the coarser sediments, the arkoses and q u a r t i z i t e s , the poorer copper with the s i l t y a r g i l ! i t e s and sha les . The geolog ica l c h a r a c t e r i s t i c s of the Copperbelt had important impl i ca t ions fo r exp lorat ion and mining: ( i ) the best areas of o r i g i na l copper depos i t ion could be located and the extensions of known bodies 73 could be predicted with f a i r accuracy. ( i i ) the ore could be separated into chunks or blocks of f a i r l y homogeneous qua l i t y so that best grades could be mined f i r s t , with some rather minor exceptions. Figure 2 depicts the map of the Copperbelt and the l oca t i on of the major depos i t s , whi le Table XIII gives the average grade of the ore mined in three Zambian mines. It can be seen that with f a i r consistency the best grades were mined f i r s t . At Roan Antelope mine, for instance, the average grade of FIGURE 2: MAP OF THE COPPERBELT AND LOCATION OF MAJOR COPPER DEPOSITS Source: Bradley [1952] -152-TABLE XIII AVERAGE GRADE ( a ) OF ORE MINED IN THREE ZAMBIAN MINES 1932-1975 YEAR Roan Antelope Muful i ra YEAR Roan Antelope Muful i ra Nkana 1932 3.65 - 1954 2.18 3.24 1933 3.61 - 1955 2.16 3.38 2.59* 1934 3.24 - 1956 2.09 3.11 2.55 1935 3.24 6.34 1957 1.95 2.86 2.53 1936 3.25 5.06 1958 1.87 2.57* 2.53 1937 3.28 4.61 1959 1.97 2.74 2.46 1938 3.26 4.53 1960 1.85 2.43 2.43 1939 3.25 4.59 1961 1.84* 2.59 2.49 1940 3.10 4.60 1962 1.81 2.77 2.34 1941 3.18 4.54 1963 1.73 2.73 2.29 1942 2.86 4.54 1964 1 .73 2.59 2.26 1943 2.53 4.09 1965 1 .80 2.55 2.11 1944 2.74 3.86 1966 1 .90 2.47 2.00 1945 2.63 3.42 1967 1.88 2.49 2.08 1946 2.59 3.41 1968 1 .79 2.50 1.97 1947 2.54 3.58 1969 1 .82 2.50 1.93 1948 2.26 3.19 1970 1 .75 2.47 1.90 1949 2.35 3.12 1971 1 .71 1 .71 1.87 1950 2.38 3.21 1972 1.55 _ 1.78 1951 2.38 3.36 1973 1 .56 _ 1952 2.40 3.43 1974 1 .48 1953 2.28 3.35 1975 _ (a) In percent *A11 f i gures below the a s te r i sk re fe r red to the average grade of ore mi l l ed (We found no f i gures on the average grade of ore mined fo r these years However, from the f r e years f o r which both f i gures were ava i l ab l e no ' s i g n i f i c a n t discrepancy was observed). Sources: Company Reports, and the Mining Yearbook of Zambia. -153-the ore mined have f a l l e n from 3.65 percent in 1932 to 1.48 percent by 1974, and at Muful i ra mine from 6.34 in 1935 to 1.71 by 1971. Est imation of (5.17) requires two time se r i e s : the cumulative ore tonnage, D^, and the cumulative copper tonnage, A^. The construct ion of these ser ies requ i res , in tu rn , data on past copper and ore production and a long h i s to ry of ore reserve estimates and corresponding average grades. In the absence of aggregate ser ies on Zambian ore production and reserves we constructed such ser ies by gathering and reassembling produc-t i o n , reserve and grade data on eleven ind iv idua l mines fo r the period 1930-1975. Several problems were encountered because of the intervening war years , d i f f e r i n g accounting pract i ces among companies and changing s t ructure and ownership of ind iv idua l mines. Data have been gathered from a number of sources such as ( i ) (Sk inner ' s ) Mining .International Yearbook ( i i ) Mining..Yearbook of Zambia ( i i i ) Company Reports ( iv ) Minerals Yearbook (v) Bancroft [1961] and Coleman [1971]. A l l tonnage f i gures were converted to metric tons fo r comparabi l i ty and cons i s tency. Figures reported according to varying accounting year ends, e i the r across mines or over time, are assumed to r e f e r simply to the 74 corresponding calendar year . Table XLVI reports the ore product ion, reserve and grade ser ies fo r ind iv idua l mines during the period 1945-1975. The aggregate ser ies on ore reserves and average grade were constructed by adding up the ore reserves of the ind iv idua l mines and taking the weighted average of t h e i r corresponding grades. The product of the two aggregate ser ies y i e l d s a th i rd s e r i e s , the aggregate copper reserves , -154-which, however, are not comparable to the f i gures on re f ined production because of i r r e d u c i b l e technolog ica l and other losses in recovery. Recovery rates vary cons iderably among mining operat ions , ranging from 60 to 90 percent. Here, we adjust the copper reserve f i gures by two a l t e r n a t i v e recovery ra tes : a high rate of 85 percent and a low rate of 70 percent. The constructed aggregate ser ies on ore reserves , average grades, copper reserves , and recoverable copper reserves are reported in Table XIV. Aggregate ore production ser ies were constructed by adding up the quant i t i e s of ore hoisted by ind iv idua l mines over the period 1945-75, reported in Table XLVI. Aggregate copper production f i gures were obtained from Table X. The cumulative ser ies on copper and ore production were obtained by cumulating the corresponding aggregate ser ies s t a r t i n g with 1944 as a benchmark year. The 1944 f i gures were, in tu rn , generated by cumulating the production f i gures s ince 1932, the f i r s t year of l a r ge - s ca le mining in Zambia. Both the annual and cumulative ser ies on ore and copper production are reported in Table XV. The cumulative ore tonnage, D^, was obtained by adding the material estimated as ore reserves at any given time to the ore produced to that time; that i s , by simply adding up the two aggregate ser ies on ore reserves and cumulative ore production constructed above (Table XIV). S i m i l a r l y the cumulative copper tonnage, A^, was constructed by adding -up the two aggregate ser ies on recoverable copper reserves and cumulative copper production given in Table XV. The cumulative ore tonnage ser ie s and two cumulative copper tonnage ser ies - one for each recovery rate - are -1 55-TABLE XIV ESTIMATED ORE RESERVES, AVERAGE GRADE, COPPER RESERVES AND RECOVER-ABLE COPPER RESERVES, ZAMBIA, 1931, 1945-1975. (Reserves: 'OOO of metric tons; grade: copper content per ton of o re ) . YEAR RESORE AVGR RESCU RES85 RES70 1945 401536.00 0 .0392 15736 20 13375.80 11015.30 1946 394740.00 0 .0391 15430 40 13115.80 10801.30 1947 389350.00 0 .0392 15243 10 12956.60 10670.10 1948 391796.00 0 .0390 15283. 90 12991.40 10698.80 1949 383768.00 0 .0390 14978. 40 12731.70 10484.90 1950 410178.00 0 .0390 15996. 90 13597.40 11197.80 1951 400998.00 0 .0389 15578. 80 13242.00 10905.10 1952 420783.00 0 .0370 15552. 10 13219.30 10886.50 1953 414675.00 0 .0369 15305. 70 13009.80 10714.00 1954 408288.00 0 .0368 15016. 80 12764.30 10511.80 1955 515814.00 0 .0372 19162. 50 162888100 13413.70 1956 550213.00 0 .0371 20434. 90 17369.70 14304.40 1957 593044.00 0 .0369 21847. 80 18570.60 15293.40 1958 603413.00 0 .0371 22410. 70 19049.10 15687.50 1959 615523.00 0 .0368 22651. 20 19253.50 15855.90 1960 624719.00 0 .0368 22977. 10 19530.60 16084.00 1961 632838.00 0 .0364 23010. 00 19558.50 16107.00 1962 623369.00 0 .0362 22541. 00 19159.90 15778.70 1963 665455.00 0 .0361 24002. 90 20402.50 16802.10 1964 658457.00 0 .0356 23427. 90 19913.70 16399.50 1965 697298.00 0 .0356 24823. 80 21100.20 17376.60 1966 690682.00 0 .0353 24367. 30 20712.20 17057.10 1967 705512.00 0 .0346 24382. 50 20725.10 17067.70 1968 693143.00 0 .0341 23698. 60 20143.80 16589.00 1969 742188,00 0 .0334 24811. 30 21089.60 1 736'7790 1970 765106.00 0 .0331 25332. 70 21532.80 17732.90 1971 796744.00 0 0329 26244. 70 22308.00 18371.30 1972 807048.00 0 .0321 25898. 20 22013.40 18128.70 1973 833530.00 0 0315 26256. 20 22317.80 18379.30 1974 851178.00 0 0311 26480. 10 22508.10 18536.10 1975 897921.00 0 0303 27180. 10 23103.10 19026.00 RESORE: AVGR: RESCU: RES85: RES70: Aggregate 6re reserves Average grade of ore reserves Aggregate copper reserves Recoverable copper reserves, assuming 0.80 recovery Recoverable copper reserves assuming 0.75 recovery Source: Estimated from data on ore reserves and average grade of i nd iv idua l mines (Table XLVI). -156-TABLE XV ORE PRODUCTION, COPPER PRODUCTION, CUMULATIVE ORE PRODUCTION AND CUMULATIVE COPPER PRODUCTION, ZAMBIA, 1944-1975. ( 'OOO of metric tons) . YEAR QORE QCU QQORE QQCU 1945 7583.48 196.92 85090.70 2415.61 1946 7381.12 185.12 92471.80 2612.52 1947 7537.20 196.01 100009.00 2797.64 1948 7708.71 216.88 107718.00 2993.65 1949 9568.96 263.19 117287.00 3210.53 1950 10309.40 281.31 127596.00 3473.68 1951] 11437.40 313.97 139034.00 3754.99 1952 11404.70 317.60 150438.00 4068.97 1953 12236.80 368.42 162675.00 4386.57 1954 14059.90 384.76 1767355000 4754.99 1955 13700.50 352.99 • 190436.00 5139.74 1956 16026.30 403.81 206462.00 5492.m 1957 17155.20 435.57 223617.00 5896.55 1958 15897.50 400.18 239515.00 6332.12 1959 20427.40 543.56 259942.00 6732.30 1960 22666.10 576.23 282608.00 7275.86 1961 21563.50 575.32 304171.00 7852.09 1962 21262.20 562.61 325434.00 8427.40 1963 22696.00 5588002 348130.00 8990.02 1964 24853.00 632.49 372983.00 9578.04 1965 27950.10 696.01 • 400933.00 10210.50 1966 25181.50 623.41 426114.00 10906.50 1967 27825.80 663.34 453940.00 11529.90 1968 30868.40 684.21 484809.00 12193.30 1969 32519.10 628.86 517328.00 12877.50 1970 31902.90 684.21 549230.00 13597.10 1971 30541.70 650.64 §79772.00 14281.30 1972 33853.00 717.79 613625.00 14931.90 1973 35012.70 706.90 648638.00 15649.70 1974 35953.70 697.82 684592.00 16356.60 1975 34312.20 676.95 718904.00 17054.40 QORE: Ore hoisted QCU: Copper produced QQORE: Cumulative ore hoisted QQCU: Cumulative copper produced Source: QORE: Table XLVI of Appendix C QCU: Table X. -157-reported in Table XVI. In estimating equation (5.17) we assume that the actual cumulative copper tonnage deviates from the "true" tonnage by an additive disturbance term e^, due to errors in measurement or aggregation over mines. Then (5.17) becomes: A t = A (1 - exp (- b.D )) + e t (5.26) Assuming further that is normally distributed we employ nonlinear 75 least squares to obtain estimates of the parameters AQ and b. Equation (5.17) was estimated for two different sets of data corresponding to the assumed alternative recovery rates. In both cases the estimated coefficients were statistically significant (even at the 0.005 level of significance) and the f i t as measured by the R statistic was satisfactory (see Table XVII) Because of the low Durbin-Watson statistic (D-W) we examine the possibility of f irst order autocorrelated stochastic structure (see Chapter 6 below). The estimates of the coefficients were altered slightly, while the D-W statistic improved substantially. Both the initial estimates and the corrected ones along with the corresponding statistics f o r both set of data are reported in Table XVIT:'. Having obtained estimates for AQ and b we can now construct a number of useful ser ies on Zambia's copper resources, to be used in the estima-tion of mining cost function and the intertemporal optimization model. In what follows we use the estimates of RQ and b corresponding to 70 per-cent recovery rate. By imposing the energy barrier at 0.001 cut-off grade, through equation (5.21) we obtain Zambia's ultimate copper resource, A , equal to 76.58 million metric tons. Substitution of A o n -158-TABLE XVI CUMULATIVE ORE TONNAGE, AND RECOVERABLE CUMULATIVE COPPER TONNAGE, ZAMBIA, 1945-1975. ( 'OOO of metric tons) . YEAR ORMT CUMT70 CUMT85 1945 486627.00 13626.10 15986.40 1946 487211.00 13598.90 15941.00 1947 489359.00 13664.20 15950.10 1948 499514.00 13909.30 16202.40 1949 501054.00 13958.30 16205.10 1950 537847.00 14954.60 17354.80 1951 540032.00 14974.60 17312.20 1952 571221.00 15271.30 17604.40 1953 577350.00 15469.10 17764.10 1954 585023.00 15650.60 17902.00 1955 706250.00 18907.40 21781.30 1956 756675.00 20306.70 23393.80 1957 816661.00 21626.10 24902.90 1958 842927.00 22418.30 25779.50 1959 875465.00 23129.80 26527.20 1960 907327.00 23936.50 27382.90 1961 937010.00 24535.40 27986.40 1962 948803.00 24755.00 28133.40 1963 1013580.00 26380.20 29980.90 1964 1031440.00 26610.70 30125.20 1965 1098230.00 28281.30 32004.50 1966 1116800.00 28586.20 32245.00 1967 1159450.00 29261.30 32919.20 1968 1177950.00 29465.50 32837.60 1969 1259520.00 30964.60 34686.00 1970 1314340.00 32014.50 35814.00 1971 1376520.00 33302.20 37238.70 1972 . 1420670.00 33777.70 37661.50 1973 1482170.00 34736.80 38676.00 1974 1535770.00 35588.00 39559.00 1975 1616820.00 36760.40 40837.60 ORMT: Cumulative ore tonnage (=RES0RE + QQORE given in Tables XIV and XV re spec t i ve l y ) CUMT85: Recoverable copper tonnage'(=RES85 + QQCU given in Tables XIV and XV re spec t i ve l y ) CUMT70: Recoverable copper tonnage (=RES70 + QQCU, where RES70 is a lso djven in Table XIV). Sources:. Computed from data given in Tables XIV and XV. -159-TABLE XVII ESTIMATION OF THE ULTIMATE COPPER RESOURCE OF ZAMBIA' Recovery Rate = 85% Parameters 0 and S t a t i s t i c s A o t - s t a t i s t i c b t - s t a t i s t i c R2 D-W P A o t - s t a t i s t i c b t - s t a t i s t i c R2 D-W P I n i t i a l Estimates 78,368.5 (33.1533) 0.461370xl0" 6 (25.4875) 0.9980 0.5439 Recovery Rate = 70% 81-628.8 (30.2242) 0.377923xl0~ 6 (24.3202) 0.9986 0.3343 Estimates Corrected for F i r s t Order. Autocorrelation 75,555.6 (19.7571) 0.487893x10"6 (14.886) 0.999.2 1.8338 (6.0874) 79,086.0 (15.5174) 0.391135xl0" 6 (12.4399) 0.9995 1.8217 0.86154 (7.8357) a. Estimating equation: (5.26); data: Table XVI b. A Q : ultimate copper resource in thousands of p : f i r s t order autocorrelat ion coe f f i c i en t . -160-fo r in equation (5.20) y i e l d s an estimate of Zambia's u l t imate ore resource, D Q , equal to 8,831.30 m i l l ion tons. Note that the to ta l f i gu re fo r DQ corresponding to A Q before the imposit ion of the energy ba r r i e r was of the order of 40,275.00 m i l l i o n tons. The int roduct ion of the c u t - o f f of 0.1 percent e l iminated 80 percent of the above f i gu re as common rock which due to i t s extremely small copper content reduced the u l t imate copper resource only by 3 percent. From equation (5.23) we obtain the unexploited port ion of the resource at each point in time, that i s , the copper resource stock, R ,^ by simply subtract ing cumulative copper production from A Q (or A Q ) . The corresponding ore resource stock at each point in time, i s obtained by subtract ing cumulat ive-ore production from the u l t imate stock D Q. The r a t i o of R^  to i s simply the average grade of a t time t'j G^ . A l l three se r ie s are reported in Table XVIII. In the same Table we report the average grade of cumulative ore production obtained from equation (5.25) using data from Table XV. -161-TABLE XVIII COPPER RESOURCE STOCK, ORE RESOURCE STOCK AND CORRESPONDING AVERAGE GRADES - ZAMBIA, 1945-1975. (Stocks: 'OOO of metric tons; grades: copper content per ton of o re ) . YEAR RCU SOR AGS AGQH 1945 74170.4 8746240.00 0.00848 0.0300 1946 73973.4 8738860.00 0.00846 0.0300 1947 73788.3 8731320.00 0.00845 0.0300 1948 73592.3 8723610.00 0.00843 0.0300 1949 73375.4 8714050.00 0.00842 0.0300 1950 73112.3 8703740.00 0.00840 0.0298 1951 72831.0 8692300.00 0.00837 0.0297 1952 72517.0 8680890.00 0.00835 0.0296 1953 72199.4 8668660.00 0.00832 0.0296 1954 71831.0 8654600.00 0.00829 0.0295 1955 71446.3 8640900.00 0.00826 0.0294 1956 71093.3 8624870.00 0.00824 0.0293 1957 70689.4 8607720.00 0.00821 0.0293 1958 70253.9 8591820.00 0.00817 0.0292 1959 69853.7 8571390.00 0.00814 0.0291 1960 69310.1 8548720.00 0.00810 0.0296 1961 68733.9 8527160.00 0.00806 0.0289 1962 68158.6 8505900.00 0.00801 0.0287 1963 67595.9 8483200.00 0.00796 0.0286 1964 67007.9 8458350.00 0.00892 0.0285 1965 66375.4 8430400.00 0.00787 0.0284 1966 65679.4 8405220.00 0.00781 0.0282 1967 65056.1 8377390.00 0.00776 0.0281 1968 64392.7 8346520.00 0.00771 0.0280 1969 63708.5 8314000.00 0.00766 0.0278 1970 62988.9 8282100.00 0.00760 0.0276 1971 62304.7 8251560.00 0.00755 0.0275 1972 61654.1 8217710.00 0.00750 0.0273 1973 60936.3 8182690.00 0.00744 0.0272 1974 60229.4 8146740.00 0.00739 0.0270 1975 59531.6 8112430.00 0.00733 0.0269 RCU: Copper Resource stock SOR: Ore resource stock AGS: Average grade of SOR AGQH: Average grade of cumulative ore hoisted Source: Computed using formulae (5.20), (5.21), (5.22), (5.23), (5.24) and (5.25), and Tables XV and XVII. FOOTNOTES 10, CHAPTER, 5 -162-1. For the presentat ion of the data we adopt the fo l lowing convention. Data constructed as part o f the present study and used in the est imation of funct ions or the so lu t ion of the model are reported in th i s chapter, while raw or intermediate data are inc luded in Appendix C. 2. The market va luat ion approach as descr ibed by C.S.O. [1972:13] c a l -culates the c a p i t a l stock of each industry as the current value of "gross f i xed assets less cumulated deprec iat ion up to date as reported on the companies' balance sheets . " 3. The ind iv idua l sectors are: Commercial A g r i c u l t u r e , Mining and Quarrying, Manufacturing, Transport and Communications, Construct ion, and Serv ices . 4. There i s some controversy over the appropriate deprec ia t ion r a te . Diewert [1977:72] reports that , on the one hand, Fe ld s te in and Rothschi ld [1974] reviewed some negative empir ica l evidence for the constant geo-metr ic rate but,.on the other hand, Hulten and Wykoff [1977] found that i t i s a good approximation to the " t r u e " rate except f o r the e a r l i e r years of the a s se t ' s l i f e . 5. Actual cap i t a l consumption allowances are expressed in current Kwachas and represent the cap i ta l deprec iat ion al lowable fo r taxat ion purposes. They are ca l cu l a ted at ce r t a in rate of the book value of the firm ' ss:;assets. However, th i s deprec iat ion rate is subject to change by tax laws and the re fo re , i t cannot be considered as a constant geometric deprec ia t ion ra te . 6. I t may be argued that the second reason does not necessar i l y imply the need fo r reconst ruct ing the ser ies on the ind iv idua l non-mining sectors rather than simply aggregating McPherson's s e r i e s . However, s ince aggregation requires the construct ion of renta l p r i c e s , which are not independent of the assumed deprec iat ion r a te , the second reason gitfen simply re in fo rces the f i r s t . 7. The perpetual inventory approach is discussed and employed by Woodland [1972b] [1975] [1977], and Donovan [1977] and others . See a l so Kendrick [1961] [1976] and Diewert [1977]. 8. See fo r instance Woodland [1977]. 9. The source National Accounts [1954, 1964, 1971] i s used as a short form for 3 d i f f e r e n t pub l i cat ions of the C.S.O.: ( i ) National Income and  Soc ia l Accounts of Northern Rhodesia 1945-53, Lusaka [1954] ( i i ) National  Accounts and Balance o f Payment of Zambia 1954-64, Lusaka [1964] and ( i i i ) National Accounts and Input-Output Tables 1971 Lusaka [1975]. 10. Note that we use McPherson's end of 1944 stocks as our beginning of 1945 s tocks. -163-11. The comparison i s meaningful only i f our f igures are lagged by one year. Our f igures <for a g r i cu l tu re are s i m i l a r to McPherson's because he used (the same) constant geometric deprec iat ion rate fo r that sec tor . 12. In machinery we include d igg ing, loading and handling equipment, earth-moving machinery, t rucks , and other s im i l a r items while plants such as m i l l s and smelters are included under s t ruc tures . Machinery, thus de f ined, i s r e l a t i v e l y easy to move and r e s e l l say to the nearby copper mines in Za i re or to gold mines in South A f r i c a . 13. Je lenc [1975:50]. 14. A zero i n t e r e s t rate was used in accumulating a l l expansionary investment to the opening date of the mine f o r two reasons: f i r s t , the time in te rva l involved was qui te short f o r a non-zero i n t e r e s t to make a s i g n i f i c a n t d i f fe rence in the r e s u l t s ; and second, what we are i n t e r -ested in obta in ing i s a proxy fo r the productive capacity of the mine represented by the accumulated rea l investment. 15. This i s not a p e r f e c t l y s a t i s f a c t o r y approach, s ince i t imposes f i xed proport ions between the two ser ies fo r the years 1945-53, but better data are simply not a v a i l a b l e . 16. The completion dates o f the major mines are as fo l lows: Roan Antelope 1932, Rhokana 1932, Mufu l i ra 1934, and Nchanga 1939. 17. See Chamber of Mines: Yearbook [1957: Table IX], [1959: Table 9] , 18. Obviously there i s an aggregation problem in adding investment expenditures in current p r i c e s , made in d i f f e r e n t years . The problem was solved by making the corresponding adjustments to the p r i ce index, so that the 1956, 1957, 1959 f igures are weighted averages of the years in which the investment was a c tua l l y made. A l t e r n a t i v e l y we could weigh the quant i t ie s by t h e i r respect ive p r i ce s . However, s ince the cap i t a l stock i s the cumulation of real investment the re su l t would have been the same. The same adjustment was made fo r the years a f fec ted by the Chambishi mine development. 19. The same approach was fol lowed by McPherson [1976a]. See footnote 9. Woodland [1972b] also employed a .10 rate of deprec iat ion fo r machinery, .08 fo r bu i ld ing s t ruc tu re s , and 0.0677 fo r engineering s t ruc tu re s . 20. The National Accounts [1954:26] report that the 1945 deprec iat ion allowance f o r Mines and Ref iner ies was 2.2 Kmn. Using the mining investment de f l a to r given in McPherson [1976a: Table Z1.5] we obtained the 1945 real deprec iat ion allowance fo r the mining sec tor . 21. These were roughly the r e l a t i v e s izes of the two stocks in the Canadian mining industry during the same per iod (see Woodland [1972b:29]). Note, however, that the share of copper in Canadian mining i s s ub s t an t i a l l y lower than in Zambian mining. -164-22. There is some scattered evidence to support the rough order of magnitude of our starting values: See Prain [1955:5] and McPherson [1976a]. 23. Lacking better information we make the conventional assumption that capital services are proportional to the stocks (see for instance Woodland [1972b]). 24. Q t + i is the expected purchase price of a unit of capital good K. Thus, the expected value of K in the next period is. (l-6)K tQ t +^; where 6 is the depreciation rate. For K = 1 the expected value of K. in period t+1 is (l-S)Q t +-j. Then Q t +^ may be referred to as the depreciated value of a unit of the capital good in period t + 1. Note that accounts for the expected capital gains (or losses). 25. The corporation tax rate is weighted by the proportion of rental income attributed to corporations, e (equal to corporation profits divided by the rental bi l l ) and the personal tax rate is weighted by O - e). 26. Diewert [1977:70]. 27. See Woodland [1972b]. 28. The proportion of depreciation allowable for taxation purposes, V. can be derived as consumption allowances divided by the product Qt-K.. .(Dn; writing equation (5.8) we make the conventional assumption that services are proportional to the stocks (see Woodland [1972:15]). 29. Ibid. 30. Woodland [1972b] derives the personal tax rate as the ratio of personal taxes to personal income. On the same lines we derive the direct tax rate as the ratio of direct.taxes to the Net National Product at factor cost (N.N.P.) minus rural subsistence consumption (or output). The data on NNP and subsistence consumption for the years 1954-68 were obtained from Simonis [1971:54]. The 1969-72 figures were constructed from data given in the Monthly Digest of Statistics, Vol. XI. August 1975 by subtracting the sum of indirect taxes net of subsidies (Table 58), consumption of fixed capital (Table 59) and rural subsistence output from Net National Income at market prices (Table 54). 31. The implicit rate of return in mining, computed according to formula (5.8) ranged between .06 and .47 during 1954-72. Professor Diewert has kindly suggested to me the alternative of using some multiple of an ex-ogenous bond rate. -165-32. The annual average y i e l d on US corporate bonds i s m u l t i p l i e d by two in order to br ing the rate of return on mining investments in Zambia to the same order of magnitude with the rate of return from mining operat ion o f comparable r i s k elsewhere e.g. Ch i l e ( fo r de t a i l s see Table XXXVIII e s p e c i a l l y the comments on RCB). 33. Both Zambian copper companies, the Anglo-American Corporation of South A f r i c a (AAC) and the Rhodesian Se lec t ion Trust (RST), borrow at the in ternat iona l cap i t a l markets and in p a r t i c u l a r at the US market. The i r cost o f borrowing was assumed somewhat higher than that of t h e i r pacenifctcompanies as a re su l t of the higher and less d i v e r s i f i e d r i s k of t h e i r operations in a developing economy. Moody's ra t ing f o r American Metal Climax (AMAX) of New York (parent company of RST) ranges between A and Baa. A- rated bonds "are considered as upper medium grade o b l i g -a t i on s . . . bu t elements may be present which suggest a s u s c e p t i b i l i t y to impairment sometime in the f u tu re . " While Baa- rated bonds "are con-s idered as medium grade ob l i ga t ions i . e . they are ne i ther highly pro-tected nor poorly secured . . . Baa- rated are more sen s i t i ve to changes in economic c ircumstances." (Moody's Bond Record Nov. 1971: 6, 60). 34. Mineral r o y a l t i e s and export taxes are treated here as i n d i r e c t taxes. Although there i s some controversy over t h e i r proper treatment (see Harvey [1971:62]) i t seems c l e a r that from the point of view of the renta l p r i ce formula (5.7) such taxes should not be included s ince they do not a f f e c t "the input p r i ces which r e f l e c t the actual costs paid by the f i rm for the use of the inputs involved in the production process" (Diewert [1977:73]). Since 1970, however, a new tax system was i n t r o d -uced imposing a 51 percent mining p r o f i t tax while the 45 percent income tax continued to be l ev ied on the remainder. The t r a n s i t i o n a l year 1970, i s t reated here according to the o l d system on the grounds that the i n -dustry had not yet made the necessary adjustments. 35. Due to the gfceat d i s p a r i t i e s in wages and s a l a r i e s between Afr icans and Non-Africans the labor s t a t i s t i c s fo r each sector are given separately under the c l a s s i f i c a t i o n s of A f r icans and Non-Afr icans, or Af r icans and Europeans. For the purpose of the present study labor i s regarded as being e i t h e r Zambian or expatr iate? with the s i m p l i f i c a t i o n of cons ider ing a l l Non-Africans employed in the mining sector as expatr iates and a l l those employed in the res t of the economy as Zambians. This could be j u s t i f i e d on several grounds: Non-Africans are only a small port ion of non-mining labor; t h e i r presence in Zambia is not assoc iated with existence of any p a r t i c u l a r industry (as in the case of the mining e x p a t r i a t e s ) ; t h e i r earnings are moderate; and they have not been the target of the 'Zambianizat ion ' p o l i c y . -166-36. Gross Domestic Product at f ac to r cost corresponds more c l o se l y to the phys ica l output than GDP at market pr ices because the l a t t e r i s a f fec ted by i n d i r e c t taxes and subs id ies . Idea l ly what we want i s a measure of the value .added by each sec tor , which i s the value of sectora l output minus mater ia l s and intermediate inputs . Unfortunately there are no data for such a refinement but i t must be kept in mind that the sectora l quant i ty ind ices (and hence our aggregate output index) " w i l l involve errors unless the r a t i o of outputs and intermediate inputs remain constant over the period"(Woodland [1972b:4Q ). 37. While the input ser ies should a lso be confined to the copper industry the Zambian s t a t i s t i c s do not permit disaggregation of the mining input ser ies between the copper and non-copper subsectors. 38. It i s true that non-copper minerals are becoming increas ing ly more important in Zambia but no s i g n i f i c a n t change in t h e i r r e l a t i v e shares ( v i s - a - v i s copper) in the value of mineral production is forseen over the model's planning hor izon. 39. Domestic copper consumption i s increas ing in the process o f Zambian i ndu s t r i a l development, but no s i g n i f i c a n t change in domestic consumption's share (usual ly around 1 percent) in tc to ta l production i s an t i c i pa ted . S tockp i l i ng on the other hand depends on the p o s s i b i l i t y fo r concerted act ion by CIPEC members to in f luence copper p r i c e s . Given the r e l a t i v e l y small share of CIPEC in world copper production (33.5 percent in 1969) and the high cost of f inanc ing and administer ing such a buf fer stock the l i k e l i h o o d fo r such act ion is rather l i m i t e d . See Takeuchi [1971:17]. 40. Short tons are converted at the rate 1 short ton=.907441 metric tons and long tons at the rate 1 long ton = 1.0163 metric tons. S ter l ings are converted at the rate £ 1 = 2 kwachas arid do l l a r s at the rate $1 = 1.40 kwachas. See Bostock and Harvey [1972: XVII]. 41. While i t seems qui te p l au s ib le that construct ion mater ia ls have been included as part o f bu i ld ings and mine development, the i nc lu s i on of imported material in the investment ser ies for machinery and equipment i s doubt fu l . It has been i n f e r r e d by cross-examining scat tered f igures on cap i ta l expenditures, stores consumed and process suppl ies in Coleman [1971], Bostock and Harvey [1972] and Hal l [1965], with the investment f igures in National Accounts, The d e f i n i t i o n of cap i t a l formation given in National Accounts [1964]"~refers to mater ia l s but i t i s not c l ea r whether they have been included in the investment ser ies or in the " i n -creases in s tocks " . In any case, expenditure on non-included mater ia l s i s l i k e l y to be sma l l , compared to the order of magnitudes of a l l other inputs . 42. Jelenc [1975:48] gives f igures fo r the years 1971-73. These shares are a lso supported by f igures on Roan Antelope mine during 1948-61- given by Coleman [1971]. -167-43. Coleman [1971| reports coal and e l e c t r i c i t y costs in three mines, Roan Antelope, and Mufu l i ra 1948-61 and Chibuluma 1956-62. These costs were as low as 4-8 percent in to ta l costs ( inc lud ing r oya l t i e s and transport co s t ) . Hall [1965:308] reports that e l e c t r i c i t y cost was as low as K6 mn out of a to ta l cost of K 126 mn. F i n a l l y , from f igures in Bostock [1972] i t was ca l cu l a ted that fuel costs fo r the years 1964-69 were as low as 1.8 percent of cost of sales de l i vered to the buyer and not more than 5 percent of cost of sales at the mine. 44. The problem of accounting fo r energy costs w i l l be considered along with t ransport costs in Chapter 8 where the a l te rna t i ve s of assuming p ropor t i ona l i t y between energy and output or energy and var iab le cap i t a l are d i scussed. 45. Mikesel l [1970:34]. 46. F l e x i b l e funct iona l forms have more free parameters f o r es t im-at ion than other more r e s t r i c t e d forms l i k e the CES or the Cobb-Douglas with the same number of inputs . 47. The aggregation over imported input was chosen over the more con-ventional aggregation over d i f f e r e n t types of labor fo r two reasons ( i ) the r e l a t i v e pr ices of the two labor inputs have been changing cons ider -ably (see Table XI);aand ( i i ) the complete intertemporal model requires d i s t i n c t i o n between imported and domestic inputs . 48. The renta l pr ices o f var iab le cap i t a l constructed in sect ion 5.2 and reported in Table VIII and the annual average earnings of expatr iates given in Table XI are used along with the quant i t ies o f the two imported inputs . 49. Note that while these cost f i gures include a l l stages of copper product ion, mining, m i l l i n g and r e f i n i n g , they do not inc lude energy and transport costs (see sect ion 8.2 below). 50. See f o r instance Scott [1967:40-46] and Schultze [1974:61-64]. 51. Lasky [1950:82]. 52. Reserve estimates are accompanied by grade estimates which are weight-ed averages of the copper content of a l l material inc luded. 53. The term was a l so used by Skinner [1977] in posing the question "Can we p red i c t commodity by commodity what ult imate y i e l d s might be, and what the future has in store fo r us?" The f i r s t ha l f of the question co r re s -ponds to what we c a l l ult imate mineral resource and the second ha l f to what we c a l l simply mineral resource (s tock) . 54. See Skinner [1977:9]. -168-55. The two terms are synonymous in the present context and are used interchangeably. " I n i t i a l endowment" i s how much copper there was in the ground before any ext rac t ion took p lace, while "u l t imate resource" i s how much copper we would have produced i f we could ex t rac t i t a l l independently of economic f a c to r s . 56. The resource c u t - o f f grade i s to be d i s t ingu i shed from the reserves (or production) c u t - o f f grades. The l a t t e r depend on economic f a c t o r s , while the former depends only on geologic f a c to r s . The resource cut -o f f grade defines the boundaries of the resource: in our case i t d i s -t inguishes ore from common rock. Professor P. Bradley suggested th i s term to underl ine the above d i s t i n c t i o n s . 57. Aluminum presents a p e c u l i a r i t y . The ore of aluminum mined at present i s bauxite. But aluminum contained in both the hydroxide com-pounds in bauxite and in s i l i c a t e minerals such as ano r th i t e , s i l l i -manite and k a o l i n i t e . Once the bauxite deposits are exhausted we must turn to s i i l i c a t e minera ls . But as Skinner [1976:262] points out: "The p r i n c i p l e of concentrat ion of aluminum minerals w i l l then be a p p l i c -able and i t w i l l be poss ib le to use common rocks as s t a r t i n g mater ia ls and to separate the anor th i te , s i l l i m a n i t e and k a o l i n i t e from other minerals in the rock. " 58. Skinner [1976:262]. 59. Concentration involves separat ing the des ired mineralsfrom unwanted material by crus ing the ore , and gathering i t s grains into a concentrate conta in ing more than 90 percent of the minera l . Unl ike concentrat ion, smelting and r e f i n i n g i s a very energy- intens ive process as i t separates the metal from the res t through d i s i n teg ra t i on of s tab le chemical com-pounds (see Skinner [1976:261]). 60. Skinner [1977] estimated that moving from ore deposits to common rock as a source of scarce metals may increase energy consumption per unit of mass of metal by factors of 100 or 1000 times today 's energy consumption. 61. Skinner [1970:266]. 62. -Lasky [1950:81]. 63. Lasky [1950:82] s ta tes : " . . . constant ly improving e f f i c i e n c y in mining and processing has made i t poss ib le to mine each year , on the average, lower-grade material than was mined the year before. As a r e -s u l t , each year (again on the average) the estimate of reserves i s based o n . . . a lower average grade. Thus over the years there is1 a bu i l d up of a record of increas ing tonnage and decreasing grade" ( for the case of Zambia see Tables XIII and XIV). - 1 6 9 -6 4 . See Draper and Klingman [ 1 9 7 2 : 1 1 5 - 1 2 9 ] . Equation ( 5 . 1 2 ) i s a l so r e f e r r e d to as ' i nverse exponent ia l ' or ' l o ga r i thmic ' func t ion . 6 5 . Lasky [ 1 9 5 0 : 8 3 ] . 6 6 . Of course, the accuracy o f the re su l t s depends on the extent to which the curve can be extrapolated beyond the data i . e . the longer the ava i l ab le ser ies the more accurate the r e s u l t s . 6 7 . In the absence of an e x p l i c i t c u t - o f f grade one would expect that D would include the e n t i r e common rock with in the given area. T h i s , how-ever, does not happen because an i m p l i c i t c u t - o f f grade is imposed by the degree of prec i s ion o f the computer software usedirim est imating equation ( 5 . 1 7 ) . In double p rec i s i on programmes the i m p l i c i t c u t - o f f grade at which convergence i s declared i s much lower than in s ing le pre-c i s i o n programmes but in both cases i s sub s t an t i a l l y lower than the energy b a r r i e r of 0.1 percent copper content. 6 8 . As i t w i l l be seen below the in t roduct ion of the energy ba r r i e r in the case o f Zambia cuts down DQ by 8 0 percent and AQ only, by 3 percent. 6 9 . De Kun [ 1 9 6 5 : 1 3 0 ] . 7 0 . See f o r instance De Kun [ 1 9 5 6 ] , Reeve [ 1 9 6 3 ] and Pel l e t t e r [ 1 9 6 4 ] . 7 1 . Reeve [ 1 9 6 3 : 4 4 ] . 7 2 . Reeve [ 1 9 6 3 : 7 7 ] . 7 3 . These are at l e a s t , the impl icat ions of the s'edimentary theory o f Copperbelt ' s o r i g i n . See P e l l e t i e r [ 1 9 6 4 : 1 8 5 ] . 7 4 . For instance f igures r e f e r r i n g to March 3 1 , 1 9 4 5 , June 3 0 , 1 9 4 5 , and December 3 1 , 1945 are a l l t reated as r e f e r r i n g to the calendar year 1 9 4 6 . 7 5 . For a de ta i l ed d iscuss ion of s tochas t i c s p e c i f i c a t i o n , est im-at ion techniques, computation a lgor i thms, and problems of au tocor re la t i on see Chapter 6 . CHAPTER 6 STOCHASTIC SPECIFICATION AND METHOD OF PARAMETER ESTIMATION In Chapter 3, we introduced two bas ic technolog ies, mining and non-ming. In Chapter 4 we s p e c i f i e d funct iona l forms f o r these technolog ies: a t rans log var i ab le cost funct ion and a t rans log production func t ion . In Chapter 5 the data required f o r est imation of the technologies were cons-Stroieted.The actual est imation and the empir ica l re su l t s w i l l be discussed in Chapter 7. The purpose of the present chapter i s to discuss the econometrics of est imation which involve s tochas t i c s p e c i f i c a t i o n f o r the est imat ing equa-t ions ( sect ion 6.2), a b r i e f desc r ip t i on of the est imation technique and computational a lgor ithm employed (sect ion 6.3) and a note on autocor re la t ion and hypothesis t e s t i ng (sect ion 6.4). 6.1 Estimating Equations and Stochast ic S p e c i f i c a t i o n In sect ion 4.1 we developed a t rans log unit va r i ab le cost funct ion (4.12) fo r the Zambian mining sector and derived the corresponding var i ab le f ac to r share equations (4.13) and (4.14). These equations form a s imultan-eous system reproduced here f o r convenience: P P P •.• a n ( j s O = a Q + ^ir\(-^-) 4 a M L ( £ n F [ ) 2 + a ^ A n Z a n ^ j ' P M 1 ? + a ^ £ n R £ n ( ^ - ) : + a^nZ + a R £ n R + ± a z z U n Z ) (4.12) 1 2 + a Z R £ n Z £ n R + ^ a R R UnR) + a £nx PM S M = a M " aML £ n(p") + a Z M £ n Z + a R M £ n R ( 4 J 3 ) -171-\ = 1 " SM (4.14) For empirical estimation the above system has to be imbedded in a stochastic framework. To this purpose, we assume that the actual unit cost and the actual shares deviate from the true unit cost and the true shares respectively by an.additive disturbance term (different for each equation) due to errors in optimization. Since S^  + S^  = 1, and, given that maximum likelihood estimates are independent of the equation dropped1 we arbitrarily omit equation (4.13) from the system (4.12) - (4.14). We assume that the vector of disturbance, = [ e c ( t ) , (t)], (where t = 1,...,T and T is the number of observations in the sample), is temporally independently and identically normally d istr i -buted with mean vector zero .arid-a--non-s.inguJiarddist-ufcbanee.covarianee^matrix. Q, . Then, w ^ = (ucu^)/T is. the maximum likelihood estimate of element * wcL °^ m 1 ; r ' ' x n> where u c and u^  are T * 1 vectors of computed residuals from the unit cost function and the labor share equation respectively, and T is the number of observations in the sample. 0, is the resulting maximum likelihood estimate of Q, l a . Thus, the system of estimating equations for the mining sector, in its final form is: P P P M 1 M 2 M c L = a Q + aM£n(p-) - ^ a M LUn p-) + a ^ n Z i i n ^ ) P M L L i 2 L (6.4)* + aR^£nRjin(p—) + a z£nZ + aR£nR + j a z z (£nZ) 1 2 t-a Z R£nZ£nR + j a R R(£nR) + aT£nT + e c -172-S L = 1 " aM + a M L £ n ( P Y } " a Z M £ n Z " a R M £ n R + eL where c L = £n(c/P L ) . Analogous stochastic specification is made for the non-mining trans-log production function (4.24) and distributive share equations (4.31) and (4.32) introduced in section 4.2 and reproduced here for convenience: £n(Y/N) = b Q + b K£n(K/N) - .5 a K N ( £ n ( K / N ) ) 2 (4.24) S K = b K - a K L £n (K/N) (4.31) S L = 1 - S K (4.32) We assume that actual output and distributive shares diverge from their cor-responding true values by additive disturbance terms due to errors in opt i -mization and/or aggregation over sectors. Optimization errors may be pre-sent because the equality of distributive shares with the corresponding out-2 put e las t i c i t ies is the result of prof it maximization. Dropping the labor share equation (4.32) we assume that the vector of disturbance e^. = . [ e Y ( t ) , e-^(t) ] is temporailTyj-independently and identical ly normally distributed with mean vector zero and a non-singular covariance mat-r ix ft whose maximum likelihood estimate is denoted by Thus, the system of estimating equations in its f inal form i s : 2 Y L = b 0 + b K £ n ( ^ - - l a K N U n + K (6.5) S K = b K - a K L £ n ( ] T ) + e K where Y^ = £n'(Y/L) . Note that the time subscripts were omitted for c lar i ty of presentation. -1 73-6.2 Est imating Technique and Computational Algorithm In est imat ing systems (6.4) and (6.5) we employ the I t e ra t i ve Ze l l ne r 3 4 E f f i c i e n t Estimator (IZEF) and the Gauss-Newton computational a lgorithm to minimize |ft.| and |ft:. . | r e spec t i ve l y . In what fol lows we r e f e r only to |ft | of system (6.4) but the same comments apply to |ft . | o f system (6.5). Given normally d i s t r i b u t e d d i s turbances, minimizing |ft.| i s equiva-lent to maximizing the log of the l i k e l i h o o d func t i on , which in concentrated form i s def ined as: ml = T^-1^ m(2* + 1) - I a \Q*\ (6.6) where T is the number of observations and n the number of equations. The i t e r a t i v e Ze l l ne r technique, a member of the c lass of minimum d i s -tance est imators , involves an i t e r a t i v e process which begins by assuming that ft i s equal to an i d e n t i t y matrix of the same dimension. It then proceeds to estimate the unknown parameters by non- l inear l ea s t squares using the 5 Gauss-Newton algorithm of non- l inear opt imiza t ion . Thus, a new estimate * of ft i s obta ined, which i s in turn used in reest imat ing the parameters v ia * general ized leas t squares (GLS). Again a new estimate of ft i s obtained which allows the process to cont inue, un t i l some p re spec i f i ed convergence c r i t e r i a are met. For the econometric work in the present study our convergence c r i t e r i o n i s that the l a rges t (proport ionate) change in the parameter values and elements of the estimated covariance matrix be less than 0.01, of t h e i r values in the preceding i t e r a t i o n . Giwen the assumption of normally d i s t r i b u t e d errors the r e s u l t i n g est imator is known to converge numerical ly to both the Maximum L i k e -l ihood Est imator and the Z e l l n e r ' s Minimum Distance Es t imator 6 and hence the parameter estimates and tes t s t a t i s t i c s are invar iant to the equation dropped. Z -174-The Z e l l n e r I te ra t i ve technique and the Gauss-Newton computational Q algorithm are programmed into the Time Series Processor (TSP) supported by the Un iver s i t y o f B r i t i s h Columbia Computing Center. As TSP i s wr i t ten in s ing le p r e c i s i o n , rounding errors may re su l t in a near ly s ingu la r matrix o f sum of squares and cross products o f i n -dependent va r i ab le s . Although the program i s capable of i nver t ing a near ly s ingu lar matrix v ia a "genera l ized inverse procedure" (by drop-ping a row and a column from the matrix) the resu l t s may be unre l i ab le i f the s i n g u l a r i t y problem per s i s t s throughout the i t e r a t i v e process. In the few cases where the problem was encountered, in order to detect whether i t was due to rounding errors or to genuine presence of c o l l i n -e a r i t y among the independent va r i ab l e s , SHAZAM (an a l t e r n a t i v e econo-9 metrics computer program was wr i t ten in double p rec i s ion ) was employed. If the s i n g u l a r i t y problem was due to rounding errors we resca led the var iab les and reran TSP, but i f , ins tead, col l i n e a r i t y was detected we dropped the of fending t e r m . ^ A l l empir ica l re su l t s reported in Chapter 7 were obtained using TSP on an IBM 370-168 computer at the Un iver s i t y of B r i t i s h Columbia Computing Center. 6.3 Problems of Autocorre la t ion and the L ike l ihood Ratio Test The c l a s s i c a l s tochas t i c s p e c i f i c a t i o n of sect ion 6.1 precludes the p o s s i b i l i t y o f intertemporal c o r r e l a t i o n or autocorre lated r e s i d -ua ls . However, qu i te low Durbin-Watson (D-W) s t a t i s t i c s were obtained in some cases (see Chapter 7) which may be in terpreted as an i n d i c a t i o n of poss ib le presence of au toco r re l a t i on , although i t i s not c l e a r what i s the prec i se meaning, o f the ind iv idua l D-W s t a t i s t i c s in simultaneous -175- . equation systems. In the absence of a single D-W s t a t i s t i c f o r the entire system we made only a limited attempt to allow for autocorrelation, by tak-ing generalized f i r s t differences and postulating the following f i r s t order autoregressive scheme11 ^ • ( t ) = p . e . ( t - l ) + u . ( t ) s t = l , . . . , T , i = 1,2 (6.7) where i s the correlation c o e f f i c i e n t between (t) and e . ( t - 1 ) , while u..(t) i s an element of u^, a 2 x 1 column vector of additive disturbances with the c l a s s i c a l properties of zero mean vector and non-singular constant covariance matrix. The method of f i r s t differences involves the transform-ation of the ori g i n a l data on dependent variables, Y . ( t ) , and independent variables, X.(t), into f i r s t differences Y^(t) - p - Y . ( t - 1 ) , etc., and set-ting up the simultaneous equation system as: Y ^ t ) - . P i Y . ( t - l ) = a ( l - P i ) + e[X.(t) - PfX - U-l) (6.8) + e . ( t ) - p i £ i ( t - l ) Substituting (6.11) i n (6.12) and rearranging terms we obtain: Y ^ t ) = a ( l - p . ) + P i Y . ( t - l ) + e[X.(t) - P i X . ( t - l ) ] + u.(t) (6.9) for i = 1 . 2 The estimation technique and computational algorithm discussed in sec-tion (6.3) can be used to estimate the parameters of the transformed equation system (6.27), including the autocorrelation coefficients p . . We f i n a l l y describe the li k e l i h o o d r a t i o test s t a t i s t i c employed in hypothesis testing and in computing system R 's. The li k e l i h o o d r a t i o i s the r a t i o of the sample maximum of the lik e l i h o o d function ( L Q ) under the null hypothesis ( H Q) to the sample maximum of the l i k e l i h o o d function (L-| ) 12 under an alternative hypothesis (H-|). The lik e l i h o o d r a t i o test s t a t i s t i c -176-i s then minus twice the logarithm of th i s l i k e l i h o o d r a t i o , which i s asymp-2 t o t i c a l l y d i s t r i b u t e d as x with degrees of freedom equal to the d i f fe rence in the number of f ree parameters under H Q and H-j . The nu l l hypothesis i s re jected i f the l i k e l i h o o d r a t i o tes t s t a t i s t i c exceeds the c r i t i c a l x value corresponding to the number of degrees of freedom (D.F) and the chosen leve l o f s i gn i f i c ance (L.S) : -2[AnL 0 - fcnl^] > XD. F U.S) (6.10) Using the l i k e l i h o o d r a t i o tes t s t a t i s t i c we te s t a number of hypo-theses r e l a t i n g to the e l a s t i c i t i e s o f s u b s t i t u t i o n , the presence of Hicks neutral techn ica l change, and f i r s t order au toco r re l a t i on . As a measure of goodness of f i t we use the conventional mu l t ip le co r -2 r e l a t i o n c o e f f i c i e n t R . In simultaneous equation systems;however, a 2 separate R i s given f o r each equation while i d e a l l y we want one measure 2 of goodness of f i t f o r the en t i re system. Such a "genera l ized R " denoted ~2 by R was suggested and used by Baxter and Craggy [1970], . Blomquist [T977],". and Khaled [1977]. I 2 i s re l a ted to the l i k e l i h o o d r a t i o tes t s t a t i s t i c of the nu l l hypothesis a l l c o e f f i c i e n t s other than the constant equal to zero in a l l equations. R2 = [1 - exp|'2(£nL Q - AnL^/n+T}] (6.11) where LQ i s the sample maximum of the l i k e l i h o o d funct ion when a l l para-meters other than the constant are constra ined to zero, L-| i s the maximum of (6.6) when a l l these parameters are included in the system at t h e i r e s t i -mated values, and T i s the number of observations in each equat ion, and n i s the number of equations in the system. ~2 The so computed R w i l l be used as the goodness of f i t s t a t i s t i c in report ing the empir ica l re su l t s in the fo l lowing chapter. -177-FOOTNOTES TO CHAPTER 6 1. See Woodland [1976: 12]. l a . Method o f computation of the res idua l s to be discussed below. 2. See sect ion 4.2. 3. See Malinvaud [1970: 294-295] and Berndt, H a l l , Hal l and Hausman [1974]. 4. A b r i e f expos i t ion of the Gauss-Newton method i s found in Malinvaud [1970: 243]. 5. Although our system cons i s ts o f l i n e a r equations the parameters o f the system as a whole are not l i n e a r in the data (dependent v a r i a b l e s ) . 6. Ruble [1968] showed formal ly that the maximum l i k e l i h o o d method and the Ze l l ne r i t e r a t i v e technique are computational ly equ iva lent . 7. See Barten [1969]. 8. H a l l , B.H., [1974], Time Ser ies Processor (TSP), Technical Paper Number 2, Harvard In s t i tu te of Economic Research, Cambridge: Harvard Univer-s i t y . 9. White, K . J . , [1977], SHAZAM, An Econometrics Computer Program Depart-ment of Economics, Un iver s i t y o f B r i t i s h Columbia. 9a. White, K . J . , [1977], SHAZAM, An Econometrics Computer Program Depart-, ment of Economics, Un ivers i ty o f B r i t i s h Columbia. SHAZAM was not em-ployed throughout s ince i t cannot yet handle non- l inear est imation and aggregation methods such as the D i v i s i a index have not ye t been program= med in SHAZAM. Thus, we employ TSP fo r a l l f i n a l resu l t s and SHAZAM only in checking intermediate r e s u l t s . 10. As a ru le 'o f fending term' was considered the independent var i ab le (or i n te r ac t i on term between two var iab les ) which was s t a t i s t i c a l l y most s i g n i f i c a n t according to the l i k e l i h o o d r a t i o tes t (see sect ion 6.3 be-low). The only exception was the case in which the one of the two c o l -l i n e a r terms was the constant; s ince the theore t i ca l basis fo r i n c l u d -ing i t into the cost or production funct ion i s not as strong as that f o r the inputs or input p r i c e s , i t was dropped even though i t might have been s l i g h t l y less i n s i g n i f i c a n t than other terms (see Models A and B in Chapter 7). 11. See Kmenta [1971: 269-297]. 12. For a r igorous presentat ion of the l i k e l i h o o d - r a t i o tes t see Malinvaud [1970: 178*184]. CHAPTER 7 EMPIRICAL RESULTS AND THEIR INTERPRETATION In th i s Chapter we report and i n te rp re t the empir ica l re su l t s ob ta in -ed through est imation of systems (6.4) and (6.5) using time ser ies const ruct -ed in Chapter 5. A number of models or var ia t ions of these systems (espec-i a l l y f o r the mining sector) were est imated. For each model we report the parameter est imates, the corresponding t - r a t i o s , the log of the l i k e l i h o o d func t i on , the Durbin-Watson (D-W) s t a t i s t i c and the degrees of freedon (D.F. ) . As a measure of goodness o f f i t we compute and report a " g e n e r a l i z -ed R " f o r each model, based on formula (6.11). Using the f i t t e d shares and formulas (4.39) - (4.50), subs t i tu t i on and p r i ce e l a s t i c i t i e s a re ' computed at each observation po in t ; the estimated parameters and computed e l a s t i c i t i e s are checked to determine whether the regu l a r i t y condi t ions on well-behaved cost and production funct ions are sa t -i s f i e d . Furthermore a number of hypotheses r e l a t i n g to input s u b s t i t u t i o n , technica l change, homogeneity and autocor re la t ion are tes ted. Sect ion 7.1 is devoted to the d i scuss ion of the empir ica l re su l t s of the est imation of system (6.4) f o r the mining sector . The empir ica l resu l t s of system (6.5) f o r the non-mining sector are reported and discussed in sect ion 7.2. 7.1 Estimated Mining Technology Parameter estimates f o r the mining technology underlying the trans log unit va r i ab le cost funct ion were obtained by est imating the system of equa-t ions (6.4) f o r the per iod 1948-69 using the data constructed in Chapter " 5. 1 The uni t va r i ab le cos t , c , i s found in Table XII along with the uni t -179-quant i ty o f A f r i can l abor , i, the uni t quant ity o f imported var i ab le input (machinery + expat r ia te l abo r ) , m, and the corresponding D i v i s i a p r i ce i n d -ex, P M- The wage rate of A f r i can labor i s found in Table XI the f i xed cap-i t a l ( s t ruc tu res ) , Z., in Table VI (column KFC3) and the copper resource stock, R, in Table XVIII (column RCU). Calendar time was used as a trend var iab le to represent exogenous Hicks-neutra l techn ica l change. Note that any re sca l i ng of these ser ies that became necessary during the est imation process i s ind ica ted under the Tables report ing parameter est imates. In examining the consistency of the cost funct ion with the underlying theory we need only show that the condit ions of monotonicity and concavity in var i ab le input pr ices and convexity in f i xed input q u a n t i t i e s , are s a t i s -f i e d ; l i n e a r homogeneity in var iab le input pr ices and symmetry are imposed 2 a p r i o r i as the theory requ i re s . The cost funct ion i s shown to be monotom-c a l l y increas ing in input pr ices i f the impl ied input l eve l s or the f i t t e d values of the cost shares are non-negative at each one of the observed sam-ple po int s . Concavity o f the unit cost funct ion i s tested at each observa-t ion point by checking whether the Hessian matrix of the second p a r t i a l de-3 r i v a t i ve s i s negative s em i -de f i n i t e , or a l t e r n a t i v e l y the eigenvalues of the e l a s t i c i t y of subs t i tu t i on matrix are non-pos i t i ve . S i m i l a r l y the unit cost funct ion is convex in f i x e d input quant i t ie s i f the matrix o f the i n -4 verse e l a s t i c i t i e s of subs t i tu t i on i s po s i t i ve semi -de f i n i t e . The presence of H icks-neutra l technica l progress, the Cobb-Douglas funct iona l s p e c i f i c a t i o n ( a ^ = 1, = 1) and a number of other hypotheses are tested using the l i k e l i h o o d r a t i o t e s t . F i n a l l y i t i s des i rab le that we te s t the parametric s t a b i l i t y o f the estimated technology by reest imat ing the system f o r two mutually exc lus ive subperiods with in our sample per iod -180-and te s t ing the hypothesis of parametric equa l i ty between these two sub-per iods. Unfortunately our l im i ted sample of 22 observations and the large number of unknown parameters (11 in the unres t r i c ted system) preclude any meaningful te s t of parametric s t a b i l i t y , f o r lack of adequate degrees of freedom. Three a l t e r n a t i v e models or var i a t ions of the system (6.4) were e s t i -mated: Model A, Model B and Model C, In model A there i s no technica l change (a = 0 ) , and the copper resource stock R. i s used to represent the two d i s t i n c t e f f e c t s o f cumulative production discussed in sect ion 4.1: the " l ea rn ing by doing" phenomenon and the cumulative de te r i o ra t i on of ore grades. As these two e f f ec t s move in opposite d i r e c t i o n s , R^  i s expected to capture the net e f f e c t of cumulative productionnon cost . Model B i s i d e n t i -ca l to Model A except f o r the i nc lu s i on of H icks-neutra l technolog ica l change (HNTC). Model C i s qu i te d i f f e r e n t from models A and B in that the copper resource stock R^ . i s replaced by the average grade of cumulative ore product ion, G^, to obtain the e f f ec t s of f a l l i n g grades alone. (HNTC i s a lso inc luded) . Correspondingly, the remainder of th i s sect ion i s organized in three subsect ions. Subsection 7.1.1 reports the empir ica l resu l t s of Model A, with spec ia l emphasis on the demand and subs t i tu t i on e l a s t i c i t i e s between var iab le inputs. Subsection 7.1.2 deals with Model B with spec ia l emphasis on the i n -verse e l a s t i c i t i e s o f subs t i tu t i on between f i xed inputs . F i n a l l y , sub-sect ion 7.1.3 presents the empir ica l resu l t s of model C with spec ia l emphasis on the inverse e l a s t i c i t i e s o f i n t e n s i t y , the inverse shadow pr ices and tech-n ica l change. For a l l three models e l a s t i c i t i e s , shares and shadow pr ices were computed at every observation point but f o r reasons of brev i ty a com--181-p lete report i s given only f o r Model C. For models A and B only se lec ted years are reported. 7.1.1 Model A: Resource Stock and No Technical Change In a f i r s t attempt to estimate Model A the 1 genera l i zeds inverse 1 pro-blem discussed in sect ion 6.2 occurred repeatedly during the i t e r a t i v e pro-5 cess i nd i ca t i n g that the ' transformed' matrix of the independent var iab les was almost s ingu la r . Although the problem disappeared during the l a s t few i t e r a t i o n s , some parameters assoc iated with the f i xed factors (see Table XX below) and the constant a Q were s t a t i s t i c a l l y i n s i g n i f i c a n t at any.reason-able leve l of s i gn i f i c ance based on ind iv idua l asymptotic t - r a t i o s . Using the l i k e l i h o o d r a t i o te s t we tested the nu l l hypothesis that the unit cost funct ion passes through the o r i g i n ( a o = 0) against the a l t e r n a t i v e hypo-thes i s o f a non-zero i n te rcept (a Q ? 0). The nu l l hypothesis could not be re jec ted even at the 0.10 leve l of s i g n i f i c a n c e . The l i k e l i h o o d r a t i o tes t s t a t i s t i c and the corresponding c r i t i c a l x are found in Table XIX. The para-meter estimates of the constra ined system (a Q set at 0) and the assoc iated s t a t i s t i c s are reported in Table XX. A l l parameter estimates were s t a t i s t i -c a l l y s i g n i f i c a n t even at the 0.01 leve l of s i g n i f i c a n c e . The f i t as measured ~2 by the computed genera l ized R (=0.9619) f o r the system was qui te s a t i s f a c t o r y and the Durbin-Watson s t a t i s t i c s , (while o f doubtful value in mu l t i va r i a te equation systems) were not unreasonably low. Before we examine the ind iv idua l e l a s t i c i t i e s i t i s important to demon-s t r a te that the estimated cost funct ion i s cons i s tent with the theory i n the sense of s a t i s f y i n g the..aforementioned regu l a r i t y cond i t ions . Monotonicity i s s a t i s f i e d throughout the sample as the ca l cu l a ted f i t t e d shares at each observat ion point turned out to be po s i t i ve as the theory requ i res . Shares f o r se lec ted years are reported in Table XXI. Concavity in va r i ab le input -182-TABLE XIX MINING VARIABLE COST FUNCTION (MODELS A AND B): LIKELIHOOD RATIO TEST STATISTICS AND CRITICAL X2VALUES Null Hypotheses Degrees of Likelihood ratio x Critical values^01' freedom test statistic .10 .025 .01 (i) Zero Inter-cept (A) 1 0.093 2.71 5.02 6.63 (ii) Cobb-Douglas in variable inputs (A) 1 6.118 2.71 5.02 6.63 ( i i i ) Cobb-Douglas in fixed inputs (A) 1 4.500 2.71 5.02 6.63 (iv) No H.N. Tech-nical Change (B) 1 9.294 2.71 5.02 6.63 (v) Cobb-Douglas in fixed Inputs (B) 1 0.549 2.71 5.02 6.63 Restrictions: (i) a Q = 0; (i i) aM|_ = 0; ( i i i ) a Z R = 0; (iv) a^ = 0 and (v) a Z R = 0. (a ) Johnston [1972:427] -183-TABLE XX MINING VARIABLE COST FUNCTION (MODEL A): PARAMETER ESTIMATES Unconstrained Constrained (a = 0) Parameter Parameter Parameter Estimate "f- r a t i o Estimate t - r a t i o a o 5,3690 (0.2985) - -a M 1.6713 (11.4945) 1.6705 (11.6648) aML -0.1415 (-2.82376) -0.1416 (-2.8429) a z -4.9781 (-0.5801) -7.1279 (-5.2689) a R 49.4739 (0.8239) 30.6538 ($577(565) aZM 0.2191 (4.8856) 0.21905 (4.9196) 3RM 1.6120 (4.8678) 1.6100 (4.9262) a z z 2.1845 (1.2735) 2.6051 (5.7030) aRR 128.3680 (1.3868) 95.813 (4.7490) a ZR -3.9169 (-0.2482) -7.7437 (-3.5196) S t a t i s t i c s S t a t i s t i c s £nL 22 .6521 22 .6096 R2 0 .8321 0 .8318 D-W1 1 .5104 1 .5973 D-W£ 0 .8166 0 .8160 D.FT 12 13 Notation M: imported inputs (p r i ce s , a D i v i s i a index of expatr ia te wages and renta l pr ices of imported machinery. See Tables V l l l r ' and XI) L: domestic labor (wages, in '000 of Kwachas) Z: mining s t ructures (quan t i t i e s , in b i l l i o n s of Kwachas, 1967 pr i ces ) R: copper resource stock (quant i t i e s , in '000 of metr ic tons) -184-pr ices was checked by c a l c u l a t i n g the f i r s t n-1 p r i nc i pa l minors o f the e l a s t i c i t y of subs t i tu t i on matrix at each observation po int . Since they a l ternatve in s ign s t a r t i n g with negat ive, the e l a s t i c i t y matrix i s negative semi -de f in i te and thus concavity i s s a t i s f i e d throughout the sample per iod. S i m i l a r l y , convexity in f ixed input quant i t ie s i s checked by c a l c u l a t i n g the n p r i nc i pa l minors o f the matrix of the inverse e l a s t i c i t i e s of sub-s t i t u t i o n , and f indinggonly po s i t i ve va lues, throughout the sample, as con-vexi ty requ i re s .^ A sample of these e l a s t i c i t i e s i s found in Table XXI. Thus, the estimated cost funct ion s a t i s f i e s a l l the propert ies required by the theory fo r a v a l i d cost f unc t i on . We now proceed to the examination of the ind iv idua l subs t i tu t i on and demand e l a s t i c i t i e s , der ived by c a l c u l a t i n g the f i t t e d shares at each data point and applying the e l a s t i c i t y forumlas of sect ion 4.3. Share and e l a s t i c i t y estimates fo r se lected years are reported in Table XXI. It must be noted, however, that a l l e l a s t i c i t y estimates are point e l a s t i c i t i e s f o r which no standard errors have been computed. This must be kept in mind when comparing d i f f e r e n t e l a s t i c i t y estimates s ince no rigorous s t a t i s t i c a l statements can be made without knowing the corresponding p r o b a b i l i t y d i s t r i -but ion. Given that each pa r t i a l e l a s t i c i t y i s a non- l inear funct ion of a l l the estimated parameters, a standard e r ro r fo r each e l a s t i c i t y cannot be dr ived from the estimated var iance-covar iance matrix of the regress ion para-meters. For an "approximate" tes t o f the s i gn i f i c ance of i nd iv idua l cross e l a s t i c i t i e s see Parks [1971:135]. The estimated A l l en e l a s t i c i t i e s of subs t i tu t i on (AES), both cross and own, are qu i te small in absolute va lue, and in genera l , less than one, and have the t h e o r e t i c a l l y co r rec t s igns: a l l point estimates of own AES are -185-TABLE XXI MINING VARIABLE COST FUNCTION (MODEL A ) ^ PREDICTED SHARES, A E S ^ , AND PRICE ELAST IC IT IES^ FOR SELECTED YEARS M 1948 1951 1956 1961 1966 1969 * 0.792 0.762 0.782 0.756 0.654 0.607 * S L 0.208 0.238 0.218 0.244 0.346 0.393 °MM -0.366 -0.686 -0.475 -0.075 -0.197 -0.263 a L L -0.531 -0.701 -0.609 -0.720 -0.707 -0.628 aML 0.139 0.219 0.170 0.233 0.374 0.406 nMM -0.029 -0.052 -0.037 -0.057 -0.129 -0.159 \ L -0.110 -0.167 -0.133 -0.176 -0.244 -0.247 \ M 0.029 0.052 0.037 0.057 0.129 0.159 0.110 0.167 0.133 0.176 0.244 0.247 Notes: (a). Model A, constra ined (a Q = 0) (g) A l l en p a r t i a l e l a s t i c i t i e s o f subs t i tu t ion (Y) Notice that nMM + nLM = 0 and nLL + nML = 0 as the theory requires f o r cost funct ions l i n e a r l y homogeneous in p r i c e s . * * Notations S M , S. : pred icted shares of imported inputs , M, and domestic l abor , L, r e spec t i ve l y , a., where i , h = M, L: e l a s t i c i t i e s of subs t i tu t i on between var i ab le inputs M and L n.^  where i , h = M, L: p a r t i a l p r i ce e l a s t i c i t i e s f o r M and L. -186-negative and cross AES p o s i t i v e . The own e l a s t i c i t y fo r the domestic input (Afr ican l a b o r ) , a^, i s higher in absolute value and more s tab le than the own e l a s t i c i t y fo r the imported input (machinery plus expatr ia te l abo r ) . While the former, ranges between -0,531 and -0.763 the l a t t e r r i se s s tead i l y over time from -0.037 to -0.263..(in absolute va lue) . The cross p a r t i a l e l a s t i c i t y n o f subs t i tu t i on between fore ign and domestic inputs , 0 ^ , i s po s i t i ve (s ince the two var iab le input case pre -cludes complementarity) and sub s t an t i a l l y less than one, ranging between 0.139 in 1948 and 0.406 in 1969. Two points are of p a r t i c u l a r i n te re s t here. F i r s t , the low value of a M L ind icates a l im i ted responsiveness of the dom-e s t i c - fore ign input r a t i o to changes in r e l a t i v e input p r i ce s . This o f fe r s some support to the widely maintained hypothesis that mining technology affords only l im i ted opportun i t ies o f subs t i tu t i on between labor and cap i ta l o r , in the context o f a developing economy, between domestic and imported inputs . For ins tance, Baldwin [1966] in a study of the Zambian copper i n -dustry, argues that : Subs t i tu t ion of labor f o r cap i t a l does occur in the [copper] industry under favorable changes in r e l a t i v e f ac to r p r i c e s . . . but from a general point o f view the subs t i tu t i on p o s s i b i l i t i e s for labor in absolute terms are comparatively s m a l l . . . copper i s a cap i t a l in tens ive industry .8 Also Ireadgold [1971] r e f e r r i n g to the Bouga inv i l le Copper Project in Papua-New Guinea noted: Although the technica l c o e f f i c i e n t s of production in the [copper] industry undoubtedly have s o m e o f l e x i b i l i t y , at l ea s t at the p l an -ning stage, engineering constra ints make i t l i k e l y that the scope f o r future f ac to r subs t i tu t i on in the Bouga inv i l le pro ject w i l l be qu i te smal l . The pro jec t i s planned and l i k e l y to remain c a p i t a l -i n tens i ve . It i s of some i n t e r e s t to compare our re su l t with those of previous s tud ies . Unfortunately there are no previous estimates of the e l a s t i c i t y of s u b s t i t u --187-t ion in copper mining between e i t he r unsk i l l ed labor and cap i t a l or domestic and fore ign inputs , derived from a f l e x i b l e funct iona l form that does not impose a p r i o r i r e s t r i c t i o n s on the s i z e or va r i a t i on of th i s e l a s t i c i t y . The c lo ses t estimate with which some rough comparison might be made i s Woodland's [1975] estimate of the e l a s t i c i t y of subs t i tu t i on between equip-ment and labor in Canadian mining using a General ized Leont ie f cost func t ion . His 1961 point e l a s t i c i t y estimate was a l so sub s tan t i a l l y less than one (0.468) but s l i g h t l y higher than ours as we might expect to be the case in a more developed economy. Woodland a lso performs a l i k e l i h o o d r a t i o tes t of the nu l l hypothesis of zero f ac to r subs t i tu t i on against the a l t e r n a t i v e of non-zero s u b s t i t u t i o n . The nu l l hypothesis was d e c i s i v e l y re jec ted . Such a tes t cannot be performed in the context of a t rans log formulat ion. Instead, we tes t the nu l l hypothesis o f unitary e l a s t i c i t y of s u b s t i t u t i o n , aML = ^ ' ( i - e - Cobb-Douglas funct iona l form in var iab le inputs) by te s t ing whether a M ^ = 0 against the a l t e r n a t i v e hypothesis that a ^ 7* 0. The nul l hypothesis was re jected at the 0.025 leve l of s i gn i f i c ance but not at the 0.01 l e v e l . The l i k e l i h o o d r a t i o tes t s t a t i s t i c i s reported in Table XIX. The above i s a very strong r e s u l t to the extent that i t v e r i f i e s Baldwin's [1966] [1966a] hypothesis of low e l a s t i c i t y of input subs t i tu t i on to which Baldwin and others have a t t r i bu ted the f a i l u r e of mining in gener-at ing l inkages with the rest of the economy and promoting sustained growth: . . . t he extent of subs t i tu t i on that i s p r o f i t a b l e over wide s h i f t s in f ac to r pr i ces i s qu i te l i m i t -e d . . . This meant that the d i r e c t economic impact o f the mineral industry on the res t of the econ-omy was s l i gh t . 9a Although the above consequences f o r growth of the l i m i t e d subs t i tu t i on poss-i b i l i t i e s in mining are widely accepted in the development l i t e r a t u r e the -188-hypothesis i t se l f ghas never been formal ly tested before. While our re su l t s are by no means conc lus ive they i n d i c a t e , at the very l e a s t , that the Baldwin hypothesis cannot be re jec ted a p r i o r i , and that fu r ther research i s needed. The second po int to note about the estimate of the e l a s t i c i t y of sub-s t i t u t i o n between domestic and fore ign inputs i s that i t r i se s s tead i l y over time (as r e l a t i v e pr ices change) e s p e c i a l l y a f t e r the l a te 1950's. This i n -creasing ease of subs t i tu t i on was a lso perceived by other wr i ter s but was never e m p i r i c a l l y tes ted . Baldwin [1966a] pointed out that "changes in the copper i ndus t ry ' s labor c o e f f i c i e n t which are s i g n i f i c a n t in percentage terms do occur, as r e l a t i v e f ac to r condit ions and pr ices change during the development p r o c e s s " . 1 ^ Also Banks [1974], r e f e r r i n g to the s p e c i f i c case of the Zambian copper indust ry , s tated "It was not unt i l 1955 or l a t e r , how-ever, that the managers of copper companies were able to perceive the o p t i -mal r a t i o n a l i z i n g involved in rep lac ing a part o f the high wage European work force by A f r i c a n s . " 1 1 We proceed now to the d iscuss ion of the p r i ce e l a s t i c i t i e s of demand for var i ab le inputs which are d i r e c t l y re l a ted to the e l a s t i c i t i e s o f sub-s t i t u t i o n through formulas (4.44) and (4.45). Let us f i r s t consider the cost-minimiz ing demands f o r domestic and fore ign inputs. Shephard's lemma t e l l s us that they co inc ide with the f i r s t p a r t i a l der i va t i ves o f the cost funct ion with respect to the respect ive input p r i c e s . This i s indeed the case. The derived demand fo r fore ign inputs , M, obtained through p a r t i a l d i f f e r e n t i a t i o n of the cost funct ion with respect to ranges from 0.1773 in 1948 to 0.1423 in 1969 compared withrthe observed-demand:of-30.1793in 1948 and 0.1200 in 1969. S i m i l a r l y the derived demand fo r domestic labor ranges -189-from 0.1500 in 1948 to 0.7168 in 1969 compared with the observed demand of 0.1680 in 1948 and 0.6984 in 1969 (see Tables XII and XI f o r corresponding units of measurement). The own pr i ce e l a s t i c i t i e s o f demand are negative as expected, but qu i te small in absolute value (see Table XXI). In p a r t i c u l a r , the demand e l a s t i c i t y f o r machinery + expatr ia te personnel, n m m,ranges from -0.02896 to -0.1595, r i s i n g over the pet idd iBut s t i l T ^ : ^ value) than the p r i ce e l a s t i c i t y o f demand for l abor , ri[_|_, which ranges between -0.1104 and -0.2465. The estimate of rij_j_ i s comparable to Woodland's 1961 po int o f estimate of -0.227 f o r the Canadian mining indus t ry . The low e l a s t i c i t y of demand fo r labor has some po l i cy impl icat ions f o r the Zambian Government in her attempt to "Zambianize" the copper industry through a wage p o l i c y : large changes i n wages may be required f o r the copper industry to become a major source of employment f o r A f r i can labor. The cross p r i ce e l a s t i c i t i e s a l so have the cor rec t s ign (pos i t i ve ) and tend to be smal ler in absolute magnitude than t h e i r own e l a s t i c i t i e s i nd i c a t i n g l i m i t e d cross p r i ce responsiveness of the var i ab le inputs . Consider now the dincome d i s t r i b u t i o n consequencesodif the estimated demand and subs t i tu t i on e l a s t i c i t i e s . From elementary economics we know that in a two-input model an increase in the serv ice p r i ce of one f ac to r r e -l a t i v e to the se rv i ce p r i ce of the other w i l l cause i t s r e l a t i v e share to increase or decrease depending on whether the e l a s t i c i t y of sub s t i tu t i on i s less than one or greater than one re spec t i ve l y . In our case the r e l a t i o n -ship between the cost share of the domestic labor and the subs t i tu t i on and demand e l a s t i c i t i e s i s given by the fo l lowing formula of the "share e l a s t i -c i t y " f o r labor: -190-kl = ( 3P[) .I ( S ^ = 6 L L " S L ( 1 - a L L } = 6 L L " S L + \ L ( 7 J ) -* where 6 ^ = 1, i s l abor ' s share in uni t cos t , i s the wage ra te , i s the A l l e n e l a s t i c i t y o f sub s t i tu t i on f o r l abor , and , i s the wage 12 e l a s t i c i t y o f demand for labor. It i s c l e a r from (7.1) that having i n -e l a s t i c demand i s not a s u f f i c i e n t condi t ion fo r labor to increase i t s share by pushing up wages. It must have a s u f f i c i e n t l y small share in the f i r s t p lace. * Given our 1961 point estimates of = -0.720, n L L = -.176 and = .244, the 1961 share e l a s t i c i t y of l abor , i s ca l cu l a ted according to formula (7.1) as: 5 L L = 1 + (-.720 - 1).244 = -.176 + (1-.244) = .580 > 0 Hence, arc increase in the wage rate by 1 percent w i l l cause l abo r ' s share to r i s e by .580 percent and the share of imported machinery and technica l personnel to f a l l , assuming, o f course, that no other change occurs. This estimate i s reasonably c l o se , t o Woodland's [1975] 1961 point estimate of \l = keeping in mind that l abor ' s share in Canadian mining is higher (.442) than in Zambia, pa r t l y because of the exc lu s i on , i n our case, o f the fore ign s k i l l e d labor. A s i m i l a r formula to (7.1) can be derived fo r £ M M and The l a t t e r (being the cross share e l a s t i c i t y ) expresses the per-centage change in the, share of input M caused by 1 percent change in the se rv i ce p r i ce o f input L, (and v ice versa) . Note that in the formula fo r 5 M L the term 6 L L i s replaced by <5ML = 0, a L L by aM|_ and nL[_ by n M L - An i n -crease in the wage rate P^ w i l l r a i se the share accruing to imported inputs M, i f and only i f a M ^ > 1, which i s c l e a r l y not the case in Zambian copper mining. Looking now at the se rv i ce p r i ce ser ies of Table XII we see that -191-A f r i can wages increased f i f t e e n f o l d over the per iod 1948-69, while the D i v i s i a p r i ce index of the aggregate imported input increased only f o u r f o l d , over the same per iod . Given the low own e l a s t i c i t y of subs t i tu t i on and de-mand for labor we would expect the A f r i can labor to have increased i t s share i n the uni t cost o f product ion. Examining the pred ic ted shares of the two inputs we observe that labor almost doubled i t s share over the per iod (from .208 in 1948 i t reached .393 by 1969) at the expense of imported machinery and fore ign labor whose share f e l l from 0.792 to 0.607 over the same per i od , as can be seen in Table XXI. Turning to the f i xed f a c t o r s , the. own inverse e l a s t i c i t i e s of sub-s t i t u t i o n were po s i t i ve as the theory requires but the cross inverse e l a s t i -c i t y between the mining s t ruc tu re s , Z, and the copper resource stock, R, had the wrong s ign f o r 7 out o f the 22 observar ions, and f l uc tua ted sub s t an t i a l l y in absolute value. The same may be sa id of the inverse p r i ce e l a s t i c i t i e s fo r these f a c to r s . Since the f i xed factors are l i k e l y to be highly c o r r e -l a ted with time, we tested fo r the presence of Hicks-neutra l techn ica l change. A l i k e l i h o o d r a t i o te s t d e c i s i v e l y re jected the nu l l hypothesis o f no techn ica l change against the a l t e r n a t i v e hypothesis o f H icks-neutra l ex-13 ogenous technica l change (HNTC), as shown in Table XIX. 7.1.2 Model B: Resource Stock and Technical Change The parameter estimates of Model A with HNTC (enter ing m u l t i p l i c a t i v e -l y ) , are reported in Table XXII along with the assoc iated tes t s t a t i s t i c s . A l l estimates were s t a t i s t i c a l l y s i g n i f i c a n t at the 0.01 leve l of s i g n i f i -cance with the notable exception of a -^ , the cross c o e f f i c i e n t between the two f i xed f a c to r s . The f i t was good and the Durbin-Watson s t a t i s t i c reason-ab le. The r e g u l a r i t y condit ions of monotonicity and concavity in va r i ab le -192-input pr ices and f i xed input quant i t ie s were s a t i s f i e d at a l l observation po int s . A l l subs t i tu t i on and demand e l a s t i c i t i e s between var i ab le inputs discussed e a r l i e r proved qui te robust as they didnot change s ign or order of magnitude when HNTC was introduced. In cont ra s t , the wide va r i a t i on in the cross inverse e l a s t i c i t y of sub s t i tu t i on a Z R between f i xed inputs was e l im inated, although the wrong sign per s i s ted for 6 of the observation po ints . Since a Z R was 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 according to i t s asymptotic t - r a t i o , we tested the nu l l hypothesis that c r Z R = 1 by t e s t -ing whether a Z R = 0 (Cobb-Douglas form in f i xed input s ) , against the a l t e r -nat ive hypothesis that a Z R f 1 or a Z R f 0. The nu l l hypothesis could not be re jected even at the .10 leve l of s i g n i f i c a n c e , (see Table XXIX).; The constrained ( a Z R = 0) system i s l a b e l l e d Model B and i t s para-meter.^ estimates are reported in Table XXII. A l l estimates were s t a t i s t i c -a l l y s i g n i f i c a n t at the .01 leve l of s i g n i f i c a n c e , on the basis of t h e i r ~2 asymptotic t - r a t i o s . The R was s a t i s f a c t o r y , the Durbin-Watson s t a t i s t i c s 1 3cL improved not i ceab ly , and the regu l a r i t y condit ions were s a t i s f i e d . The demand and subs t i tu t i on e l a s t i c i t i e s (Table XXIII) had the r i gh t s ign across the sample and in magnitude were su rp r i s i n g l y c lose to those obtained from Model A without technica l change (Table XXI) and hence our e a r l i e r comments on these e l a s t i c i t i e s continue to apply. The own inverse e l a s t i c i t i e s .of subs t i tu t i on had the r i gh t s ign (pos i t i ve ) at a l l but two observations in the case o f a z z and one observat ion in the case of a R R , and they were not 1 ' unreasonab le ' in s i z e . For instance a z z was c lose to 1 fo r most observation po int s . The cross inverse e l a s t i c i t y of s u b s t i t u t i o n , a Z R , was of course, unitary throughout the sample s ince aZR = ®' Unitary e l a s t i c i t y o f sub s t i tu t i on between mining s t ruc tu re s , Z, -193-TABLE XXII MINING VARIABLE COST FUNCTION (MODEL B): PARAMETER ESTIMATES Unconstrained ^ Cftn'strained (a 7 R =0) Parameter Parameter Parameter Estimate t - r a t i o Estimate t - r a t i o aM 1.6630 (11.5722) 1.6681 (11.6692) aML -0.1522 (-3.5361) -0.1399 (-2.9383) a z -6.3268 (-5.3902) -5.2664 (-7.1303) a R 36.7880 (4.7987) 34.4272 (7.3229) aZM -0.2266 (5.4892) 0.2176 (5.0343) aRM 1.5619 (5.0216) 1.6090 (5.0007) azz -3.1860 (-7.6737) -3.0377 (-7.3700) aRR 100.6780 (6.0074) 86.5106) (6.2056) aZR -2.5789 (-0.9699) - -a_ -0.4984 (-2.9827) -0.5414 (-4.7555) S t a t i s t i c s S t a t i s t i c s Jinl 27 .2566 26 .9820 R2 0.8638 0 .8621 D-W-j 1 .5845 1 .7660 D-W2 0 .7913 0 .8204 D.F. 12 13 (a) Note that the r e s t r i c t i o n a = 0 on Model A is maintained in Model B. o Notat ion: M: imported inputs ( p r i ce s , D i v i s i a index, see notat ion to l a b l e XX L: domestic labor (wages, in '000 of Kwachas) Z: mining s t ructures (quan t i t i e s , in b i l l i o n s of Kwachas, 1967 pr i ce s ) R: copper resource stock (quant i t i e s , in '000 of metr ic tons) T : time ( in '00 years from 0.04 to 0.25). -194-TABLE XXIII MINING VARIABLE COST FUNCTION (MODEL B ) ^ SHARES, SHADOW PRICES, AND ELASTICITIES FOR SELECTED YEARS 1948 1951 1956 1961 1966 1969 * 0.792 0.762 0.781 0.756 0.655 0.608 0.208 0.238 0.219 0.244 0.345 0.392 CTMM -0.399 -0.719 -0.507 -0.785 -0.201 -0.267 °LL -0.576 -0.733 -0.647 -0.750 -0.723 -0.640 aML 0.152 0.230 0.181 0.242 0.381 0.413 nMM -0.316 -0.547 -0.396 -0.593 -0.132 -0.162 nLL -0.120 -0.175 -0.142 -0.183 -0.250V • -0.251 nML 0.316 0.547 0.396 0.593 0.132 0.162 \ M 0.120 0.175 0.396 0.593 0.132 0.162 "Z 2.296 2.057 0.776 -0.417 -0.586 -1.090 0.958 0.988 1.570 0.583 -0.614 -1.893 aZZ 0.615 0.545 -0.659 2.957 2.601 1.882 aRR 0.898 0.881 0.827 0.568 1.503 1.201 aZR 1.000 1.000 1.000 1.000 1.000 1.000 aMZ 1.106 1.130 1.462 0.436 0.468 0.684 aLZ 0.597 0.585 -0.651 2.742 2.009 1.490 aMR 1.208 1.251 1.357 1.920 -0.237 0.468 °LR 0.209 0.200 -0.276 -1.842 3.345 1.824 (a) Model B constrained (a^R =0, a Q = 0) * where i = M, L: predicted shares of variable input in unit cost where i , h = M, inputs L: Allen elasticities of substitution between var i-^ where i , h = M, L: partial price elasticities of derived demands (for variable inputs u i where j = Z, R: inverse static shadow prices of fixed inputs o.^ where j , k = Z, R: inverse elasticities of substitution between fixed J inputs c r . j . where i = M, L and j = Z, R: inverse elasticities of intensity between fixed and'lv.a'rlabilelUitfputs. - 1 9 5 -and the resource stock, R, impl ies a change 6y 1 percent in the r a t i o of t h e i r shadow pr i ce s (u>z/u)R) in response to a change by 1 percent in the r a t i o o f t h e i r quant i t i e s (Z/R). (See sect ion 4.1 f o r i n te rp re ta t i on of the shadow pr ices o f the f i xed input s ) . The inverse e l a s t i c i t i e s of i n ten s i t y expressing the r e l a t i v e e f f e c t of a change in the quant i ty o f f i xed inputs on the demand fo r var i ab le i n -puts were genera l ly po s i t i ve with the exception of fo r the years 1954-63 . The e l a s t i c i t i e s of i n ten s i t y between machinery, on the one hand, and s t ructures and the resource stock on the other (CTm z and a M R ) tend to be somewhat greater than one in e a r l i e r years and somewhat less than one in l a t e r year s . The reverse i s true f o r the e l a s t i c i t i e s between labor L, and 14 s t ruc tu re s , Z. This could roughly be in terpreted as fo l lows: a 1 percent increase in one of the f i xed inputs would have reduced the demand f o r im-ported machinery (domestic labor) by more ( less ) than 1 percent in the e a r l i e r years and by less (more) than .1 percent in the l a t e r years . The above i n t e r p r e t a t i o n , however, i s not genera l ly true fo r the re l a t i on sh ip between A f r i can l abor ,L , and the resource stock. The e l a s t i c -i t y , a ^ R , i s c lose to zero f o r the f i r s t 6 years and becomes inc reas ing ly negative therea f te r (unt i l 1963) i nd i ca t i ng a r e l a t i on sh ip of 'complemen-t a r i t y ' , between L and R. Only f o r the l a s t s i x observations i s there a c l e a r re l a t i on sh ip of ' s u b s t i t u t a b i l i t y ' ( a ^ R > 0 ) . ^ The in terprea t ion of th i s behavior o f i s to be found in the ro le at R in the cost funct ion as 1 c r e f l e c t e d in i t s inverse shadow p r i c e , u>R. F i r s t , r e c a l l that the copper resource stock, R^ ., i s the reverse of the cumulative output (see equation 5 . 2 3 ) , and i t s ro le in the cost funct ion i s to capture two d i s t i n c t e f f e c t s : ( i ) the ' l ea rn ing by doing ' phenomenon, -196-and ( i i ) the e f f e c t of the cumulative de ter io ra t i on of ore grades. As the two e f f ec t s work in opposite d i rec t ions and ' l ea rn ing by doing ' i s l i k e l y to be stronger in the e a r l i e r year s , while the e f f e c t o f f a l l i n g grades i s stronger in l a t e r yea r s , we would expect that the l a t t e r w i l l eventual ly more than o f f s e t the former. That th i s i s indeed the case can be seen by examining the behavior o f the inverse, shadow p r i c e , co R, o f the resource stock, R, over the sample per iod , obtained by p a r t i a l l y d i f f e r e n t i a t i n g the cost funct ion with respect to R. Over the subperiod 1948-56, when A f r i can labor was s t i l l inexperienced and high grade ore was s t i l l a v a i l a b l e , <JJr was inc reas ing ly p o s i t i v e . As the ' l ea rn ing by doing ' e f f e c t l eve led o f f and the high grade ore was exhausted, u>R f e l l s t ead i l y to become negative in 1963 and continued to be inc reas ing ly negative therea f te r (see Table XXIIf). The p o s i t i v e a>R in the e a r l i e r years may be thought of as the p r i ce which producers would gave been w i l l i n g to pay to get r i d of one add i t iona l unit of the resource, as th i s would have lowered t h e i r cost by an equal amount through the net l e a r n -ing e f f e c t . A negative u>R, in the l a t e r years ind icates how much producers would have been w i l l i n g to pay to acquire an add i t iona l unit o f the resource as th i s would have saved them the cost (= u R) of resor t ing to lower grade ore. Given th i s behavior of <JJr and i t s i n t e r p r e t a t i o n , i t should not be su rp r i s i ng that (which i s normalized by O J r ) i nd icates that L and R are 'complements' over the subperiod 1948-63 and ' s ub s t i t u te s ' t h e r e a f t e r . ^ 7 7.1.3 Model C: Average Ore Grade and Technical Changed To obtain separately the e f f e c t of the cumulative de te r i o ra t i on of grades we reestimated the system of equations (6.4) a f t e r rep lac ing the -197-copper resource stock, R ,^ by the average grade of cumulative ore produc-t i o n , G ^ ; 1 8 the e s t i m a t e s 1 9 of a z and a ^ were s t a t i s t i c a l l y i n s i g n i f i c a n t and were set equal to zero. The l i k e l i h o o d r a t i o t e s t s t a t i s t i c s were 1.466 f o r a^ and 1.159 f o r a R R while the c r i t i c a l Chi-square value i s x-j (.1) = 2.71. The parameter estimates of the constra ined ( a z = 0, a^g = 0) system are reported in Table XXIV. The regu l a r i t y condit ions were a l l s a t i s f i e d at a l l observation points as the var iab le input shares were po s i t i ve f o r monotonicity and the matrix o f the e l a s t i c i t i e s o f sub s t i tu t i on between va r i ab le inputs negative-semi d e f i n i t e f o r concavity and between f i xed inputs po s i t i ve semi -de f in i te f o r convexity (see Tables XXV and XXVI). A l l 9 f ree parameters were s t a t i s t i c a l l y s i g n i f i c a n t even at the 0.005 ~2 leve l o f s i g n i f i c a n c e , based on t h e i r asymptotic t - r a t i o n s , and the R was very s a t i s f a c t o r y . The f i t t e d shares and the estimated subs t i tu t i on and demand e l a s t i c -i t i e s between var iab le inputs (see Table XXV and XXVII) had cons i s ten t l y the r i gh t s ign and were of the same order of magnitude as in Models A and B. The inverse e l a s t i c i t i e s o f subs t i tu t i on and i n t e n s i t y were a lso of the same s ign and order of magnitude as in model B, except f o r those prev ious ly 20 invo lv ing R ahd now G, (see Table XXVI). The own inverse e l a s t i c i t y of sub s t i tu t i on of ore grade, o^, was very c lose to one and extremely s tab le over time. The cross inverse e l a s t i c i t y of subs t i tu t i on between the average grade of ore hoisted and mining s t ruc tu re s , a^Q, was a lso c lose to one during the subperiod 1948-58 but less than one during 1959-69; (see Table XXVI). The inverse e l a s t i c i t y o f i n ten s i t y between machinery and grades, aMg, was p o s i t i v e , less than one and extremely s tab le over time ranging between 0.86 21 and 0.89. This can be loose ly in te rpre ted as i nd i ca t i ng that 1.0 percent -198-TABLE XXIV MINING VARIABLE COST FUNCTION (MODEL C): PARAMETER ESTIMATES Parameter Parameter Estimate t - r a t i o s S t a t i s t i c s a o 47.2146 (4.8260) £nL 27.4366 aM -2.3465 (-3.3629) R2 0.8649 aML -0.1511 (-3.2020) D-W1 1.8076 a G -46.9622 (-5.1302) D-W2 0.7927 aZM 0.2275 (5.2912) D.F. 13.00 aGM 3.2318 (4.8866) a z z -3.3999 (-7.3390) a ZG -5.3057 (-6.9019) a T -0.2010 (-5.7520) Note: Parameters a z and a^g were s t a t i s t i c a l l y i n s i g n i f i c a n t even at the .10 l eve l o f s i g n i f i c a n c e and were dropped (see t e x t ) . The shares shadow p r i c e s , and e l a s t i c i t i e s of Model C reported below in Tables XXV, XXVI and XXVII were computed a f t e r the r e s t r i c t i o n s a z = 0 and aGG = ^ w e r e ' ' m P o s e c ' -Notat ion: M: imported inputs ( p r i ce s , D i v i s i a index (see notat ion to Table XX). L: domestic labor (wages in '000 of Kwachas) Z: mining s t ructures (quant i t ies in b i l l i o n s o f Kwachas in 1967 p r i ces ) G: average grade o f cumulative ore production (percent) T : time ( in years : 4 -25) . -199-TABLE XXV MINING VARIABLE COST FUNCTION (MODEL C): ALLEN ELASTICITIES OF SUBSTITUTION (a) AND PRICE ELASTICITIES ( n ) YEAR 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 MM -0.021 -0.029 -0.046 -0.054 -0.037 -0.063 -0.043 -0.049 -0.032 -0.032 -0.051 -0.039 -0.050 -0.058 -0.078 -0.092 -0.143 -0.180 -0.174 -0.234 -0.220 -0.236 LL -0.309 -0.385 -0.506 -0.541 -0.445 -0.576 -0.489 -0.520 -0.409 -0.410 -0.528 -0.462 -0.524 -0.558 -0.616 -0.638 -0.650 -0.625 -0.629 -0.570 -0.585 -0.568 ML 0.081 0.106 0.153 0.170 0.128 0.190 0.145 0.160 0.114 0.114 0.114 0.164 0.134 0.162 0.180 0.219 0.305 0.335 0.331 0.365 0.359 0.366 (a) \L -0.064 -0.083 -0.118 -0.130 -0.092 -0.143 -0.112 -0.122 -0.089 -0.089 -0.125 -0.104 -0.124 -0.136 -0.161 -0.175 -0.208 -0.218 -0.217 -0.223 -0.222 -0.223 (a) \ M 0.017 0.023 0.036 0.041 0.028 0.047 0.334 0.376 0.249 0.250 0.388 0.302 0.382 0.438 0.574 0.665 0.977 0.117 0.114 0.143 0.137 0.143 (a) We ca l cu l a ted four p r i ce e l a s t i c i t i e s , \ L ' a n c * nMl_' We report only T I ^ and r\^\ but s ince the cost funct ion i s homogeneous of degree one in pr ices and the demand funct ions homogeneous of degree zero in pr i ces and are obtained as: N M M = " \ M A N D N M L = " \ L -200-TABLE XXVI MINING VARIABLE COST FUNCTION (MODEL C): INVERSE ELASTICITIES OF SUBSTITUTION^ AND OF INTENSITY YEAR 4^  (7 rt a ( 3 ) a ( 3 ) ZG °MZ °LZ MG LG 1948 1.059 1.096 0.632 0.864 1.521 1949 1.061 1.100 0.634 0.864 1.496 1950 1.063 1.109 0.641 0.864 1.450 1951 1.066 1.117 0.630 0.865 1.430 1952 1.071 1.236 0.569 0.868 1.460 1953 1.078 1.146 0.557 0.869 1.398 1954 1.094 1.174 0.415 0.874 1.423 1955 1.105 1.199 0.354 0.875 1.406 1956 1.177 1.343 -0.227 0.883 1.421 1957 1.400 1.799 -1.861 0.886 1.407 1958 1.722 2.502 -3.843 0.886 1.369 1959 0.600 0.154 3.908 0.891 1.375 1960 0.578 0.095 3.929 0.889 1.359 1961 0.542 0.018 4.O'70O 0.888 1.349 1962 0.507 -0.094 4.079 0.885 1.323 1963 0.450 -0.241 4.274 0.883 1.308 1964 0.426 -0.401 3.979 0.877 1.261 1965 0.661 0.120 2.641 0.874 1.234 1966 0.561 -0.119 3.126 0.873 1.241 1967 0.702 0.162 2.306 0.867 1.207 1968 0.755 0.325 2.100 0.868 1.214 1969 0.857 0.532 1.728 0.869 1.204 (a) The own inverse e l a s t i c i t y of subs t i tu t ion , i s v i r t u a l l y constant over time at 1.03 while a z z i s very unstable ranging from.1.26 in 1948 to 10.6 in 1969 ( 3 ) The formulas for a Z g, a^g and are analogous to t'he'trifdwmuTas' for °ZR' °MR' -201-change in the grade of ore creates the need of about .85 percent increase in machinery to produce one unit of output. Finally the elasticity of inten-sity between African labor and grade of ore was also positive, greater than one, and steadily declining from 1.52 in 1948 to 1.20 in 1969 (see Table XXVI) reflecting possibly the improvement of labor skil ls over time. The uniformity of sign and stability over time of all elasticities associated with G are reflections of its shadow price co^  which was uniform-ly negative and steadily rising over time in absolute value. With COQ -2 2 3c/3G < 0 and 3c /3G > 0 the continual decline in grades is translated at an increasing rate into higher unit cost of production (see Table XXVII). Again, producers would have been willing to pay uto to keep the current G .b from falling to a lower level in the succeeding period. It must be kept in mind, however, that the cumulative deterioration in grades is the direct result of cumulative production, not of the passage of time. The inverse shadow price of structures, co2 = 3c / 3 Z , also follows a clear pattern identical in Models B and C: over the first eleven years (1948-58) it is positive and falling, while over the eleven years (1959-69) i t is negative and rising over time in absolute value. (See Tables XXIII and XXVII). This pattern reflects the excess capacity of the earlier years, as the structures for the four major mines (built during 1930-39) were designed with an eye towards the future. Underutilized structures meant higher unit operating cost (at the margin in 3 c / 3 Z > 0) but as the excess capacity became a binding constraint their effect on cost was reversed ( 3 c / 3 Z > 0). This is also reflected in the own inverse price elasticity of structures, n^i* w n i c h 1 S invariant to scaling, and gives the percentage change in their shadow price w 7, due to a percentage change in their quantity. -202-TABLE XXVII MINING VARIABLE COST FUNCTION (MODEL C): PREDICTED SHARES AND INVERSE STATIC SHADOW PRICES YEAR * M * S L 1948 0.7926 0.2074 2.6443 -0.7268 1949 0.7846 0.2153 2.4295 -0.7148 1950 0.7675 0.2325 2.2482 -0.7481 1951 0.7606 0.2393 2.3944 -0.8993 1952 0.7771 0.2229 2.4481 -1.0724 1953 0.7518 0.2482 2.4780 -1.3921 1954 0.7705 0.2295 2.1320 -1.6741 1955 0.7648 0.2351 1.9562 -1.8743 1956 0.7818 0.2182 1.0980 -2.3329 1957 0.7818 0.2182 0.4006 -2.3608 1958 0.7633 0.2367 0.1975 -2.2820 1959 0.7747 0.2252 -0.2811 -2.2485 1960 0.7641 0.2359 -0.2727 -2.3120 1961 0.7565 0.2435 -0.2504 -2.3027 1962 0.7378 0.2622 -0.2225 -2.2177 1963 0.7252 0.2748 -0.1930 -2.1472 1964 0.6802 0.3198 -0.1908 -2.2681 1965 0.6510 0.3490 -0.2980 -2.2765 1966 0.6551 0.3449 -0.2929 -2.8185 1967 0.6092 0.3908 -0.4782 -3.4013 1968 0.6196 0.3804 -0.5495 -3.3534 1969 0.6085 0.3915 -0.7678 -3.5253 (a) the formula for CJQ is analogous to the formula for u ) R and obtained by replacing R by G in equation (4.19). -203-It was positive during 1948-58 and negative during 1958-69, and so was the cross inverse price e l a s t i c i t y n Z R 5 indicating the percentage change in the shadow price of structures due to a percentage change in the grade of the ore. Needless to say that both the own and the cross price e l a s t i c i t i e s of the grade, H Q Q and n^, were negative throughout as i t s shadow price, wg, was negative throughout the sample period. A f i n a l comment refers to technical change. The c o e f f i c i e n t a^ . associated with time was s t a t i s t i e a l l y s i g n i f i c a n t even at the 0.005 l e v e l , and bore a negative sign in both models B and C r e f l e c t i n g the cost dimin-utive effect of Hicks-neutral exogenous technical change. The importance of technical change in mining has long been recognized in the l i t e r a t u r e . Banks [1974], for instance, wrote:L It has been said that the lowering of the average copper content of ores during the l a s t seventy years has been responsible for about 50 percent of the price r i s e of copper. In fact were i t not for the technical progress in ore mining, dressing concentrating and some metal-l u r g i c a l processes this percentage would probably be higher, and, ... the price of copper higher 22 In addition to this general form, technical progress in Zambian copper mining took two s p e c i f i c forms: ( i ) the gradual s h i f t from 'underground' methods to 'open p i t ' techniques as large open p i t mines are less costly to operate than underground mines (see Appendix A for details) and ( i i ) the i n -troduction of the TORCO process which permits profitable treatment of d i f -f i c u l t ref racatory ores and wastes (see Appendix A). In conclusion, the proposed and estimated translog variable cost function s a t i s f i e d a l l the r e s t r i c t i o n s implied by the theory of cost mini-mizing behavior. The fixed inputs were shown to contribute to the explana% tory power of the function and had generally the expected effect on unit -204-var iab le cos t s . The f i t t e d shares, e l a s t i c i t i e s o f sub s t i tu t i on and i n t e n -s i t y had genera l ly the r i gh t s ign and were not invar i an t to r e l a t i v e p r i ce changes. Thus, our resu l t s do not j u s t i f y the use of r e s t r i c t e d funct iona l forms that e i t h e r suppress the response of the input shares to p r i ce changes (Cobb-Douglas) or require the e l a s t i c i t i e s of subs t i tu t i on to be constant (CES). Although,=there ex i s t s no s ing le c r i t e r i o n whereby the r e l a t i v e per-formance of the estimated models can be judged, o v e r a l l , Model C appeared to perform bet ter than Models A and B, intterms of s a t i s f y i n g the regu la r -22a i t y condit ions and y i e l d i n g ' reasonable ' e l a s t i c i t i e s in some loose sense. 7.2 Estimated Non-mining Technology Given the high l eve l o f aggregation of the non-mining sector and the l im i ted time ser ie s a v a i l a b l e , only l im i ted a t tent ion i s paid to modeling and est imat ing the non-mining technology. Parameter estimates were obtained by est imat ing the system (6.5) dertve'd from ancaggregate t rans log production func t i on , that excludes mining. The sample per iod i s that of the years 1954-72 f o r which non-mining data were construoteddin Chapter 5. The aggregate ser ies on non-mining output, c a p i t a l and labor and the correspond-ing D i v i s i a p r i ce ind ices are found in Table IX?.. In add i t ion calendar time i s used as a trend var i ab le to represent Hicks-neutra l exogenous technica l change (HNTC). In the present sect ion we report the parameter est imates, correspond-ing te s t s t a t i s t i c s , and the 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 . The monotonicity and concavity of the production funct ion in input pr ices was checked and a number of hypotheses r e l a t i n g to au toco r re l a t i on , technica l change, and sub-s t i t u t i o n were tes ted. -205-System (6.5) was estimated with both exponential and multiplicative HNTC, but as the results were basically the same, we report only the former. Parameter estimates, asymptotic t-ratios? D-W statistics, the log of l i ke l i -hood function and the generalized R are found in Table XXVIII. The f i t was satisfactory and all parameter estimates were statistically significant even at the .005 level of significance. However, as the D-W statistics were quite low in both equations of the system we tested the null hypothesis of no autocorrelation against the alternative hypothesis of diagonal f irst order autoregressive structure. The null hypothesis was decisively re-jected. The likelihood ratio test statistic was 10.671 while the critical 2 value of x for two degrees of freedom at the .0] level of significance is 9.210. Hence, all further discussion assumes the presence of diagonal f irst order autoregressive structure (see section 6.3). Table XXVIII reports the parameter estimates and associated statistics of the complete system assuming the presence of f irst order autocorrelation. Again, all coefficient estimates were statistically significant even at the .005 level based on the corresponding asumptotic t-ratios. Both D-W ~2 statistics- and the R improved substantially in comparison to their values under the classical stochastic specification. The esti-mated translog production function was monotonically increasing in input quantities as the fitted distributive shares were strictly positive throughout the sample period (see Table XXIX). Furthermore, i t was concave in input quantities as the elasticity of substitution matrix proved to be negative-semi definite. Thus, all regularity conditions on well-behaved production functions were satisfied. Using the fitted shares and formulas (4.49) and (4.50) we computed -206-TABLE XXVIII NON-MINING PRODUCTION FUNCTION PARAMETER ESTIMATES Classical Stochastic Specification First-Order Auto-correlation (F.O.A.) rameter Parameter Estimate t-ratio Parameter Estimate t-ratio bo -0.5069 (-12.8017) -0.5111 (-6.5492) bK 0.5206 (44.9268) 0.5218 (29.0267) bKN -0.1522 (-6.1383) -0.1589 (-4.1616) b T 0.0284 (9.3881) 0.0283 (5.0259) P l - 0.5538 (2.9489) P 2 - 0.4684 (2.4151) Statistics Statistics m l 10.5014 15 .8367 R2 0.6334 0 .7422 D-W] 0.8645 1 .8571 D-W2 0.9327 1 .4336 D.F. 14 12 Notation: K: non-mining capital (Divisia index; see Table IX) N: non-mining labor (Divisia index; see Table IX) T : time (in years: 2-19) p : first-order autocorrelation coefficient 1,2: subscripts referring to the production function and capital share respectively. -207-and reported in Table XXX the own and cross e las t ic i t ies of substitution at each data point. Al l e las t ic i t ies had the correct sign, the own being negative and the cross, positive at each observation point. The cross e las t ic i ty of substitution between capital and labor, a ^ , was substan? i a l l y less than one (around 0.35) and surprisingly stable over time. As the e las t i c i ty of substitution between mining labor and imported machinery computed in the previous section was of the same approximate mag-nitude, our findings offer l i t t l e support to an implicit hypothesis in a 23 large part of the development l i terature that the substitution poss ib i l -i t ies are more limited in mining than in non-mining technology. This comparison, however, must be regarded with caution as the translog variable 24 cost function is not dual to a translog production function. Further-more, the s tat i s t ica l significance of these e las t ic i t ies has not been test-ed, although, a l ikelihood ratio test rejected the null hypothesis of unitary e las t ic i ty of substitution, a ^ , between labor and capital against the alternative hypothesis of f 1. The l ikelihood ratio test s ta t i s t i c was 6.027 compared with a c r i t i c a l x value of 3.841 at the 0.05 level of significance. F inal ly, the null hypothesis of no technical change (b T = 0) was tested against the alternative of Hicks-neutral exogenous technical change (b^ f 0). The null hypothesis was rejected at the 0.05 level of significance but not at a higher level . The likelihood ratio test s ta t i s t i c was 4.938; for 2 c r i t i c a l x values with 1 degree of freedom see Table XIX. Some interesting points may be made by examining Table XXIX. F i r s t , the share of capita l , S^, has increased over time at the expense of the labor share, S N, as we would expect to be the case in a developing economy, -208-TABLE XXIX NON-MINING PRODUCTION FUNCTION (F.O.A.)^: PREDICTED SHARES AND MARGINAL PRODUCTS YEAR * Si/ * SM K N *K $N 1955 0.3601 0.6399 0.4037 0.2593 1956 0.3759 0.6241 0.4072 0.2699 1957 0.3924 0.6076 0.4101 0.2813 1958 0.4042 0.5958 0.4155 0.2922 1959 0.4282 0.5718 0.4146 0.3072 1960 0.4396 0.5604 0.4205 0.3195 1961 0.4506 0.5494 0.4266 0.3323 (1962 0.4566 0.5434 0.4356 0.3440 1963 0.4605 0.5395 0.4459 0.3553 1964 0.4700 0.5300 0.4548 0.369.1 1965 0.4820 0.5180 0.4598 0.3847 1966 0.5001 0.4999 0.4631 0.4039 1967 0.5230 0.4770 0.4642 0.4268 1968 0.5419 0.4581 0.4681 0.4492 1969 0.5606 0.4394 0.4725 0.4729 1970 0.5729 0.4271 0.4804 0.4940 1971 0.5793 0.4207 0.4912 0.5123 1972 0.5839 0.4161 0.5031 0.5302 (a) With f irst order autoregressive (stochastic) structure. * Notation: S„: predicted share of non-mining capital K; predicted share of non-mining labor N; marginal product (static shadow price) of K; marginal product of labor N. rN -209-TABLE XXX NON-MINING PRODUCTION FUNCTION (F.O.A.)^ ALLEN ELASTICITIES OF SUBSTITUTION AND PRICE ELASTICITIES YEAR aKK aNN CTKN \N nNK 1955 -0.552 -0.175 0.311 -0.111 0.199 1956 -0.536 -0.194 0.323 -0.121 0.201 1957 -0.517 -0.216 0.334 -0.131 0.203 1958 -0.502 -0.231 0.340 -0.138 0.203 1959 -0.468 90.263 0.351 -0.150 0.201 1960 -0.453 -0.279 0.355 -0.156 0.199 1961 -0.437 -0.294 0.358 -0.161 0.197 1962 -0.428 -0.302 0.360 -0.164 0.195 1963 -0.422 -0.308 0.361 -0.166 0.195 1964 -0.408. -0.321 0.362 -0.170 0.192 1965 -0.391 v -0.338 0.364 -0.175 0.188 1966 90.364 -0.365 0.365 -0.182 0.182 1967 -0.331 -0.398 0.363 -0.190 0.173 1968 -0.304 -0.426 0.360 -0.195 0.165 1969 -0.278 -0.453 0.355 -0.199 0.156 1970 -0.262 -0.470 0.350 -0.201 0.150 1971 -0.253 -0.479 0.348 -0.202 0.146 1972 -0.247 -0.486 0.346 -0.202 0.144 (a) With f irst order autoregressive (stochasti c) structure Notation: o„„, am: own Allen elasticities of substitution (AES) between non-mining capital, K, and labor N, respectively a.,..: cross AES between K and N NN' nKN : partial price elasticities. Note that because of homogeneity of «(•): nKK = -nNK and nKN = -nNN . -210-starting with very l i t t le capital. Second, the marginal product of labor, grows faster than the marginal product of capital, $ K > as a result of the capital deepening that takes place during the development process. In conclusion, our results do not justify the use of the Cobb-Douglas functional form in representing the non-mining production technology since input shares are not invariant to price changes or alternatively the hypo-thesis of unitary elasticity is rejected. However, a less restrictive form, the CES, is not precluded as our estimate of is virtually constant throughout the sample period. Having estimated the technological parameters we turn to Chapter 8 to investigate how these estimates might be used in obtaining a numerical solution to the intertemporal model of Chapter 3. -211-FOOTNOTES TO CHAPTER 7 1. Note that although data were constructed f o r the per iod 1945-70, in actual est imation we employ only the f i gures fo r the years 1948-69. The f i r s t few years were excluded from the sample s ince the con-s t ructed stock f igures f o r these years are l i k e l y to be qu i te s e n s i -t i v e to deprec iat ion assumptions and benchmark stocks. The l a s t observat ion was a l so excluded from the sample s ince T970 was a year of t r a n s i t i o n from the o ld system of taxat ion and fore ign ownership to the new per iod of n a t i o n a l i z a t i o n . 2. However,theireeexiistia'formal tes t of l i n e a r homogeneity in input pr ices and symmetry. The former i s d i r e c t l y testab le through equa l i ty con-• s t r a i n t s ( i ) ' - ( v ) ' on equation (4.10). S i m i l a r l y symmetry may be tes ted by est imat ing the system with and without the symmetry cond i -t ions a M L = a L M and a Z R = a R z ; . 3. This i s done by ca l cu l a t i n g . the values of the p r i n c i p a l minors fo r each equation and checking i f the f i r s t n-1 ordered p r i nc i pa l minors a l te rna te in sign commencing with negat ive. The nth order p r i nc i pa l minor w i l l be zero because of the assumed homogeneity in var iab le inputs . Note, however, that t h i s does not cons t i tu te a s t a t i s t i c a l t e s t of concavity s ince the s t a t i s t i c a l s i gn i f i c ance of the p r i nc i pa l minors has not been determined. 4. While convexity in f i xed input quant i t i e s i s not as c r u c i a l as con-cav i t y in var i ab le input p r i c e s , i t should be S a t i s f i e d in most of the observations f o r a well-behaved var i ab le cost func t ion . 5. i . e . transformed by a 'weight ing ' matr ix, derived from ft (see sect ion 6.2 above). 6. In the two-var iab le - input cases with l i n e a r homogeneity in p r i c e s , i t i s necessary and s u f f i c i e n t f o r concavity that the own A l l en p a r t i a l e l a s t i c i t i e s o f sub s t i tu t i on are negative at a l l observation po ints . See a lso footnote 3 above. 7. Since no homogeneity in f i xed inputs i s imposed we had to ca l cu l a te a l l n p r i n c i p a l minors. Note that convexity was v i o l a ted in one observat ion, 1967. 8. Baldwin [1966:80]. 9. Treadgold [1971:196]. 9a. Baldwin [1966a:51]. Note that Baldwin emphasized a l so the presence of economies of sca le in the production of mining inputs and some other technolog ica l c h a r a c t e r i s t i c s of the production funct ion (see Baldwin [1966]). 10. Baldwin [1966:81] This quote must be in terpreted with caut ion. It i s only re levant to the extent that whatever changes take place during the development process are expressed in terms of changing r e l a t i v e -212-p r i ce s . The estimated cost funct ion i s a s t a t i c re l a t i on sh ip which cannot capture dynamic " s t ruc tu ra l change"; but because of the f l e x i b i l i t y o f the funct iona l form the e l a s t i c i t y of subs t i tu t i on changes as the r e l a t i v e pr ices change. 11. Banks [1974:83], It i s not c l e a r from th i s quote (or the text from which i t was taken) whether the lack of subs t i tu t ion between imported and domestic labor (before 1955) was due to errors in opt imizat ion on the part of the management or to a low e l a s t i c i t y of subs t i tu t i on at the p r e v a i l i n g (before 1955) r e l a t i v e input p r i ce s . The quote i s , at l e a s t , i n d i c a t i v e of the concern o f e a r l i e r wr i ters with the problem. We found, however, no previous attempts to estimate the mining production technology f o r Zambia. 12. Formula (7.1) i s the r e su l t o f p a r t i a l d i f f e r e n t i a t i o n of the labor share equation (4.14) with respect to P^ and the property that can be wr i t ten i n terms o f the p a r t i a l der iva t i ves of the un i t cost funct ion (see formula (4.36)). For de t a i l s see Woodland [1975:179-1801]. The impl i ca t ions of (7.1) have been derived independently by Sato and Koizumi [1973:484-489] and Woodland [1972a]. 13. Note that a was s t a t i s t i c a l l y i n s i g n i f i c a n t even a f t e r the i n t r o -duction of .HNTC-. Thus, a l l d iscuss ions and Tables on Model B assume a = 0. o 13a. Convexity in var iab le input pr ices was s a t i s f i e d throughout the sample whi le the concavity in f i xed input quant i t ies was s a t i s f i e d in a l l but the f i r s t few observat ions. 14. T p i s i - i s a a n i n t u i t i v e l y appealing but ' l oo se ' i n te rp re ta t i on of the e l a s t i c i t i e s o f i n t e n s i t y . Consider, f o r ins tance, the formal i n t e r p r e t -itationon of the e l a s t i c i t y o f i n ten s i t y between machinery and the r e -source stock: = r e l a t i v e change in (M/UR) = d(M/"R) ,M/"R , CTMR r e l a t i v e change in (R/P M) d(R/P M) ' R / P M ' gives the percentage changes in the r a t i o o f demand fo r imported machinery to the shadow pr i ce of the resource due to 1 percent change in the r a t i o of the quant i ty o f the resource to the se rv i ce p r i ce of the imported machinery. Thus, e i t he r a f a l l in R (given P^) or a r i s e i n P M (given R) or some combination of the two w i l l cause the r a t i o (M/'dL) to f a l l . Under th i s q u a l i f i c a t i o n we w i l l continue to i n t e r -pret the e l a s t i c i t i e s o f i n ten s i t y in a somewhat loose sense i m p l i c -i t l y assuming no change in var i ab le input p r i c e s . 15. The use of the terms 'complementarity' and ' s u b s t i t u t a b i l i t y ' i s sub-j e c t to the above (footnote 14) q u a l i f i c a t i o n s 16. See sect ion 4.1 f o r the meaning of the inverse shadow pr ices (pp. 104-105). -213-17. See footnote 14 above. 18. The average grade of cumulative ore product ion, G., has been defined H in equation (5.24). There i t was denoted by G. but here f o r s i m p l i -c i t y we omit the superscr ip t H. Note that given the assumptions of sect ion 5.5 and the d e f i n i t i o n of R^ . (equation 5.23) i t can be shown that G. i s an increas ing funct ion of R.. The G. time ser ies i s given i n Table XVIII. z z 19. Two add i t iona l modi f icat ions were made and should be kept in mind when comparing the re su l t s o f Model C with those of Models A and.B. F i r s t , the constant was s t a t i s t i c a l l y s i g n i f i c a n t and was re introduced. Second, the m u l t i p l i c a t i v e HNTC term (Model B) was replaced by an exponential HNTC as the former was col l i n e a r with G. 20. The inverse e l a s t i c i t i e s of subs t i tu t i on between var iab le inputs and mining s t ructures were.,general lycposit ive^exGeptffo ' r i thrreeooMerva'-" t ions (1956, 1957 and 1958) in the case of a.7 and four observations (1962, 1963, 1964 and 1966) in the case of a j ^ . 21. See footnote 14 above. 22. Banks [1974:69-70]. 22a. Unusually large (or small) or w i l d l y f l u c tua t i n g e l a s t i c i t i e s may be viewed as 'unreasonable 1 . 23. See f o r instance Baldwin [1966], [1966a], Reynolds [1965] and Gomez-D'Angelo [1973]. 24. The t rans log funct iona l form i s not s e l f - d u a l . I f the production funct ion i s t rans log then the cost funct ion w i l l not be t rans log and v ice versa. While the production model postulates that the technology can approximately be descr ibed by a t rans log funct iona l form the cost model assumes that there e x i s t some production funct ion of unknown form whose subs t i tu t i on e l a s t i c i t i e s are adequately represented by the parameters o f a t rans log cost func t ion . CHAPTER 8 TOWARDS A NUMERICAL SOLUTION While a f u l l numerical so lu t ion of the planning model developed in Chapter 3 i s beyond the scope of the present study i t i s of some i n te re s t to examine how the estimated production technologies and constructed cap i t a l stocks might be used in obta in ing such a so lu t ion and what add i t iona l information might be requ i red. A numerical so lu t ion involves t rac ing numerical ly the optimal time paths of the p o l i c y instruments (x,v,q,w) and the corresponding t r a j e c t o r i e s of the cap i t a l and resource stocks ( k ,z , r ) over the planning h o r i z o n . 1 The importance of computing optimal p o l i c i e s and studying the features of optimal paths i s t h ree fo l d . F i r s t , a numerical so lu t ion may help to judge the past performance of actual economies and determine whether past p o l i c i e s were conducive to optimal resource u t i l i z a t i o n . Second, i t provides a way of est imat ing the optimal rate of resource ext rac t ion and the optimal consumption-investment a l l o c a t i o n f o r a government which i s both a resource owner and a development planner. T h i r d , i t provides a means of est imat ing 'account ing ' pr ices which can be used as a guide in e s t ab l i sh ing a system of taxat ion in a decent ra l i zed economy. The estimated production technologies provide us with values of the s t ruc tura l parameters of the model and the estimated cap i t a l stocks cons t i tu te the i n i t i a l resource cond i t ions . In add i t i on , a f u l l numerical so lu t ion requires ( i ) a d i s c re te s p e c i f i c a t i o n of the soc ia l welfare func t i on , ( i i ) long-run project ions of pr ices and other exogenous parameters, J = e" p tu(x t)dt (3.7) - 215 -( i i i ) a discrete time approximation of the entire continuous time planning model, and (iv) an appropriate solution algorithm. The present chapter is devoted to a brief discussion of these points. 8.1 The Social Welfare Function In Chapter 3 we introduced a continuous time intertemporal social welfare function of the form: r T . 0 where x is the per capita consumption of the homogeneous commodity y,'.T is the planning horizon, e " p t a discount factor and u(-) a momentary uti l ity function, which is strictly concave and monotonically increasing in x. Since economic data come in discrete forms, empirical applications require a discrete time approximation to the above continuous time social welfare function. Consider the following discrete approximation to ( 3 . 7 ) : T . J° = I (1 + p ) _ t u(x t) (8.1) t=l 1 where t represents a time period (say a year), T is a time horizon (say twenty years) and u(-) is a one-period uti l ity function. Equation (8.1) is an additively separable (cardinal) social welfare function which rules out any possible complementarities of consumption enjoyed at different periods. For empirical applications we need, further, to specify a functional form for the one-period uti l i ty function u(x). Any functional specifica-tion that preserves the monotonicity and concavity of u(-) in x is admissible. In models of optimal growth the most widely used form is the isoelastic uti l ity function: - 216 -u(x) = x 1 _ n 0 < n < l ; o r l < n < » (8.2) where n i s a constant parameter. The i s o e l a s t i c u t i l i t y funct ion was f i r s t introduced in optimal growth models by Ramsey [1928] and subsequently used by Fr i s ch [1932], Tinbergen [1960], Chakravarty [1969], Kendrick and Taylor [1971], Dasgupta and Heal [1974], Vajed [1975] and others . Chakravarty [1969] extens ive ly studied the propert ies of the i s o e l a s t i c u t i l i t y func t ion : ( i ) u = x n > 0 :: monotonicity of u(-) in x _ i ( i i ) u = - nx~ < 0 : d iminishing marginal u t i l i t y (MU) du x x u x ( i i i ) ? = -r— • — = = 1 - n : (constant) e l a s t i c i t y of u t i l i t y du xu ( iv ) e = - - j — — = = n : (constant) e l a s t i c i t y of marginal x u x u x u t i l i t y with respect to consumption (v) £im u(x) = x : l i n e a r i t y at the l i m i t (as n-K) . approaches zero) The choice of the s p e c i f i c funct iona l form (8.2) i s usua l ly j u s t i f i e d on grounds of ana l y t i c a l convenience. Chakravarty [1969] provided a fu r ther j u s t i f i c a t i o n by po int ing out that the constant e l a s t i c i t y n "can be regarded as a very convenient index of the equa l i t a t i an bias [among 4 generat ions] that we want to impart to our p lann ing. " A higher value fo r n means a sharper f a l l in the marginal u t i l i t y of consumption as the l a t t e r r i s e s over t ime. Subst i tut ing equation (8.2) into equation (8.1) the d i s c re te time-soc ia l welfare funct ion becomes: - 217 -J° = I 0 + P ) _ t T ^ x ] - n . (8.1a) t=l n There remains the question of how we can obtain estimates of the subject ive parameters p and n . UNI DO |1972| provided some p rac t i c a l suggestions on how to conduct " Socra t i c dia logues" with po l i c y makers to e l i c i t the soc i a l ra te of time preference, p , and the e l a s t i c i t y of the marginal u t i l i t y of consumption, n . However, as Manne [1974] pointed out " i n th i s pos t - Socra t i c era one can subs t i tu te computer time for dialogues and can take refuge in s e n s i t i v i t y a n a l y s i s - t e s t i n g which numerical values lead to p l aus ib le time paths for the system. Once [ p ] and [ n ] have been s p e c i f i e d , there i s no d i f f i c u l t y in the a n a l y s i s " . The s e n s i t i v i t y ana lys i s need not be conf ined to the numerical values of p and n - A l t e rna t i ve funct iona l s p e c i f i c a t i o n s of u(-) can a lso be attempted. It must be noted, however, that the funct iona l form of (8.2) i s not as r e s t r i c t i v e as might appear at f i r s t s i ght . A great number of d i f f e r e n t funct iona l forms preserving monotonicity and concavity a re , in f a c t , reduced to a constant e l a s t i c i t y of u t i l i t y funct ion when appl ied to the single-consumption-good case. Cons ider, f o r ins tance, some of the (add i t ive homothetic) u t i l i t y funct ions , that belong to the 'Bergson fami ly (see Po l lak [1971:402]) n u(x) = 1 b^ in x i b i > 0 u(x) = - I b .x a b. > 0 i=l 1 1 u(x) = I b .x a b. > 0 i=l 1 1 ^ = 1 (8.3) i a < 0 (8.4) 0 < a < 1 (8.5) - 218 -where x = (x-j . X g , . . • ,x ). Equation (8.3) i s of the "Cobb-Douglas" type while (8.5) i s Houthakker's [1960] d i r e c t addi log u t i l i t y func t i on . The ind i f fe rence maps of the u t i l i t y funct ions (8.3) , (8.4) and (8.5) are i den t i ca l with the isoquant maps of the c o n s t a n t - e l a s t i c i t y - o f - s u b s t i t u t i o n production funct ions (see Chipman [1965]). In the one-good case (i = 1), however, the u t i l i t y funct ion does not represent an i nd i f f e rence map but, s imply, the transformation of uni ts of consumption into uni ts of u t i l i t y . The one-consumption-good vers ions of (8.3) - (8.5) are as fo l lows: u(x) = b Jin x b > 0 (8.3a) u(x) = - bx a b > 0 a < 0 (8.4a) u(x) = bx a b > 0 0 < a < 1 (8.5a) Equations (8.4a) and (8.5a) are constant e l a s t i c i t y of u t i l i t y funct ions : with s = a and e = a - 1 where a < 0 f o r (8.4a) and 0 < a < 1 fo r (8.5a). The u t i l i t y funct ion (8.3a) has a l so constant e l a s t i c i t y of marginal u t i l i t y with respect to consumption'(e = -1) but the e l a s t i c i t y of u t i l i t y i s not constant (? = x/in x ) . In t h i s sense (8.3a) i s a somewhat more general funct iona l form than the other three. Carter [1967] employs equation (8.3a) while Margl in [1967] uses a modif ied vers ion of (8.4a). A more general funct iona l form for u(-) may be obtained from the t rans log s p e c i f i c a t i o n proposed by Chr i s tensen, Jorgenson, and Lau [1972]. The t rans log d i r e c t u t i l i t y funct ion may be wr i t ten as: n n n an u(x) = I a- in x- + h I I b.. Jin x. in x. . (8.6) i=l i=l j=l 1 J J Equation (5.6) i s analogous to the t rans log production funct ion (see sect ion 4.2) and i s assumed to s a t i s f y the propert ies of monotonicity - 219 -and concavity in x ^ 7 The one commodity ( i =1 ) vers ion of (8.6) may be wr i t ten as fo l lows: 2 in u = a in x + h b (£n x) (8.6a) Q Equation (8.6a) i s assumed to have the fo l lowing p roper t ie s : ( i ) u v = u(a + b an x)/x > 0 : monotonicity ( i i ) u v v = - (u v /x ) + (ub/x 2 ) < 0 : d iminishing MU XX X X ( i i i ) s = xu /u = a + b an x X ( iv ) e = xu /u =-1 + a + b £ n x + b/(a + b an x ) . : XX X From ( i i i ) and ( iv ) i t i s obvious that ne i ther the e l a s t i c i t y of u t i l i t y , c , nor the e l a s t i c i t y of the marginal u t i l i t y with respect to consumption,. e , are constant. Both ? and e depend on the leve l of consumption x. If, however, homotheticity (b = 0) i s imposed, equation (8.6a) becomes a constant e l a s t i c i t y u t i l i t y funct ion with u v = au/x, u = (u / x ) ( a - l ) , X XX X t. = a and e = a -1, where 0 < a < 1. (Notice the s i m i l a r i t y with (8.5a)). We propose that in numerical s imulat ions, equations (8.2) , (8.3a), and (8.6a) be attempted as a l t e rna t i ve funct iona l s p e c i f i c a t i o n s of u(x) in order to tes t the s e n s i t i v i t y of the numerical re su l t s to ( i ) the constancy of the e l a s t i c i t y of u t i l i t y alone and ( i i ) the constancy of both the e l a s t i c i t y of u t i l i t y and the e l a s t i c i t y of MU with respect to consumption. The f i n a l comment i s reserved f o r the assumption of add i t i ve separab-i l i t y over time embodied in the intertemporal soc i a l welfare funct ion (8.1). In modell ing the intertemporal cho ice , the optimal growth l i t e r a t u r e (s ince Ramsey [1928]) r e l i e d on a preference funct ion which depends - 220 -a d d i t i v e l y on consumption at the various dates within the planning hor izon. Although Hicks ' [1965] view that there are strong complementarities between consumption in successive periods i s widely he ld, i t i s seldom modelled "because we do not know how to spec i fy [ intertemporal complementar-i t i e s ] in an a n a l y t i c a l manner which would be deemed adequate to the 9 problem." Ryder and Heal [1973] made a f i r s t attempt to model optimal growth with in ter tempora l ly dependent preferences by postu la t ing that instantaneous (or s ing le -per iod ) s a t i s f a c t i o n depends not only on instantaneoud consumption but a l so on the customary or expected leve l of consumption. The l a t t e r i s introduced in the one-period u t i l i t y funct ion through a new var iab le which i s a weighted average of past consumption l eve l s with the weights dec l in ing exponent ia l l y into the p a s t . 1 ^ Ryder and Heal ' s experiment ind icates that re lax ing the assumption of add i t i ve s e p a r a b i l i t y complicates immensely the s t ructure of optimal growth models. It appears that f o r some time to come add i t i ve s e p a r a b i l i t y over time w i l l continue to be r e l i e d upon, in the absence of "any s ing le commanding hypothesis to take i t s p l a c e . " 1 1  8.2 Long-run Pr ice Project ions Our planning model involves three pr i ce va r i ab le s : ( i ) copper p r i c e s , p, ( i i ) imported mining input p r i c e s , g, and ( i i i ) non-mining import p r i c e s , p y . In chapter three, fo r s i m p l i c i t y , p Y was assumed constant and equal to one while the other two pr ices were treated as g iven. While i t i s not u n r e a l i s t i c to assume that these pr ices are exogenously g iven, we must not ignore the f ac t that they w i l l be changing over the planning hor izon. Since copper pr ices f l uc tua te cons iderably in the short-run (as - 221 -noted in Chapter 2) , the re levant pr ice var i ab le in long-term planning i s a range of expected export pr ices over a twenty or t h i r t y year per iod. The pro ject ion period (which must co inc ide with the planning hor izon) - should be long enough to cover several normal recess ions and booms, the t iming of which cannot be pro jected. S i m i l a r l y , the planners need long-run project ions of import pr ices (g and Py) s ince i t i s not the absolute leve l of copper pr ices that determines the resource benef i t s to the country but the r e l a t i v e pr ices or terms of t rade. As the Central S t a t i s t i c a l O f f i c e of Zambia states in the National Accounts: "The quant i ty of goods and serv ices that can be purchased every year from abroad using copper export earnings would cons t i tu te a more re levant ser ies . . . than . . . f i gures i nd i ca t i ng the 12 quant i ty of production of copper." Here, however, we focus on copper pr i ce project ions s ince import pr ices in very open economies l i k e Zambia, fo l low c l o s e l y the general world p r i ce leve l which may be eas ie r to fo recas t . There e x i s t a number of econometric models engaged in mineral p r i ce project ions but t h e i r p red i c t i ve power decreases r ap id l y with the length of the time per iod , and "by t h e i r very nature are not su i tab le f o r long-run 13 f o reca s t i n g . " A rather t yp i ca l example i s the well-known econometric forecas t ing model f o r copper developed by F i sher , Cootner and Ba i l y [1972]. Evaluat ing t h e i r project ions the authors state "the f i r s t th ing to note about these forecas t s i s that they are undoubtedly better as one year 14 project ions than as forecasts over several year s . " Ana lys i s of long-run world demand project ions in r e l a t i o n to project ions of world mine capac i ty and mining cost trends throughout the world provide the basis for pred ic t ions with regard to the minimum and - 222 -maximum long-run average p r i ce s . For any expected leve l of demand, mining costs tend to set a f l o o r on copper pr ices whi le costs f o r new mines i n the lower-cost producing areas, and the costs of development and use of subst i tutes tend to set a c e i l i n g on the long-run average p r i ce of copper. In p a r t i c u l a r , the p r i ce of aluminum which i s the c lo ses t subs t i tu te fo r copper sets an upper l i m i t to the long-run pr i ce of copper. While i t i s not our in tent ion to engage in long-run pr ice p ro jec t i ons , a number of re levant impl i ca t ions of the theory of non-renewable resources may help to i l l u s t r a t e how such project ions may be made. The exhaust ib le 15 resource model t e l l s us tha t , given a homogeneous resource, the pr ice of the metal in the ground or user;cost of the resource (which i s equal to the market pr i ce of the metal minus ex t rac t ion cost ) i s r i s i n g over time 16a a t the market rate of d i scount, assuming, of course, that the quant i t i e s demanded at various pr ices do not change over time: where p i s the world market pr i ce of the minera l , c i s the world average ex t rac t ion cos t , and v i s the world market discount r a te . If ext ract ion costs were zero the market pr ice of the metal would be r i s i n g at the rate of i n t e r e s t : But s ince ext rac t ion costs are not zero the r e l a t i v e rate of r i s e in the market pr i ce of metal w i l l be below the rate of d iscount. With pos i t i ve but constant ex t rac t ion cost ( i . e . , c ° + ^ = c°) the time p r o f i l e of p may be represented by the d i f fe rence equat ion: D t + 1 " c ? + l = 0 + v ) [p t - c° ] (8.7) P t + 1 = (1 + v )p t (8.7a) - 223 -P t + 1 = P t + v ( p t - c°) (8.7b) In the case of a homogeneous resource and no technica l change i t i s not u n r e a l i s t i c to assume f i xed ext rac t ion costs i f the serv ice pr ices of mining inputs are expected to remain f a i r l y constant over the time horizon considered. Introducing resource heterogeneity, exogenous technica l change and input pr i ce changes, we may s t i l l r e ta in the assumption of f i xed ext rac t ion costs i f the e f f e c t s on costs of these fac tors can be reasonably expected to o f f s e t each other. In genera l , however, ex t rac t ion costs w i l l vary over time and correspondingly p^+^ w i l l depend on, both c t and c^ + - j : P t + 1 = 0 + v ) [ P t " c°] + c ° + ] (8.7c) Since c^+-j i s not known the problem of p red i c t ing long-run pr ices i s replaced by the problem of p red ic t ing long-run cos t s . Assuming that mining input pr ices fo l low the general world pr i ce leve l (assumed constant) and that exogenous technica l change lowers costs at an exponential r a te , d, then: pt+1 = (l+v)pt - ( v+d )A /R° (8.7d) where A/R° i s average ex t rac t ion co s t s , c ° , (so that the parameter A determines i n i t i a l average cost) and R° i s the world resource stock of the given mineral ( in terms of metal content) . Average cos t s , thus, approach i n f i n i t y as the resource base approaches zero. Equation (8.7d) gives the metal p r i ce of next year as a funct ion of the p r i ce of the current year , of the current resource stock and of two parameters, the rate of d i scount, and the rate of technica l change. Pindyck [1976] through an optimal mining model f o r exhaust ib le resources with no - 224 -technica l change (d = 0) a r r i ved at a formula fo r p ^ s im i l a r to (8.7d): P t + 1 = (1 + v ) p t - vA/R° (8.7e) Applying equation (8.7e) to the case of copper and drawing on the econometric model by F isher Cootner and Ba i ley [1972] and the study by Banks [1974] Pindyck derived copper p r i ce t r a j e c t o r i e s fo r the years 1975-2010 under 18 a l t e rna t i ve assumptions regarding d iscount ing and copper market s t ruc ture . The above ana lys i s was conducted under the assumption that the quant i t i e s demanded at various p r i ces do not change ( i . e . , demand curves do not s h i f t ) . What i f changes in tastes and technology, growth of populat ion and discovery of new sources ( inc lud ing new subst i tutes ) cause a shift in the demand for copper? In such a case a cons iderably more general se t t i ng such as a model of the world economy i s required f o r the determina-t ion of the long-run pr i ce of copper in r e l a t i o n to a l l other long-run p r i c e s . Leont ie f [1977] develops such a model of the world economy and attempts long-run pr i ce pro ject ions under a l t e rna t i ve assumptions regarding resource a v a i l a b i l i t y , population growth, and technolog ica l developments: . . . there w i l l be a tremendous growth in the world consumption of minerals between 1970 and 2000. The demand for copper i s expected to increase 4.8 t imes, f o r bauxite and z inc 4.2 t imes, n ickel 4.3 times . . . These estimates were adjusted wherever poss ib le to take into account the in f luence of future techno l -ogies on resource development and consumption . . . and the f a c t that . . . as more access ib le reserves of p a r t i c u l a r minerals become exhausted, the next layer invo lv ing higher ext rac t ion costs begins to be exp lo i ted 19 Leont ie f [1977] der ives projected mineral (and other) pr i ces for .the years 1980, 1990 and 2000 under pess imi s t i c and op t im i s t i c estimates of world 20 resource endowments: " A v a i l a b i l i t y of l a rger high-grade deposits postpones increases i n ex t rac t ion costs with cumulative output . . . and impl ies lower - 225 -21 pr ices for resources" L e o n t i e f ' s projected pr i ces are 'normal ized ' so that the average pr i ce of a l l goods consumed would remain the same as i t was in 1970. Having ind ica ted how long-run p r i ce project ions might be made in presenting a d i s c re te s p e c i f i c a t i o n of the planning model, and a so lut ion a lgor i thm, in sect ion 8.3 and 8.4, we w i l l f o r s i m p l i c i t y be using the fo l lowing exponential approximation of equation (8.7d): P t + 1 = 0 + e ) P t where e < v (8.7f) The pr ices g and o , of imported mining inputs , m, and the imported commodity Y, r e s p e c t i v e l y , w i l l be assumed constant to maintain comparab-i l i t y with the theore t i ca l model of chapter 3. In p r i n c i p l e , changing import pr ices could be .ea s i l y accommodated in the model. Before concluding th i s s ec t i on , r e c a l l that the uni t va r i ab le cost of copper ex t rac t ion was defined as being net of energy and transport costs because'of lack of data. Transport costs may be assumed proport ional to output and deducted from the world copper pr i ce to obtain the Zambian minesite copper p r i c e . If un i t t ransport costs fo l low the general world p r i ce leve l (assumed constant ) , no add i t iona l compl icat ion i s introduced; otherwise long-run project ions of t ransport costs are a l so necessary. The s i t ua t i on with energy costs i s somewhat d i f f e r e n t . F i r s t , energy costs are l i k e l y to be re l a ted to cap i t a l inputs ( in our case machinery and s t ructures ) rather than output. Second, to the extent that energy i s obtained from non-renewable sources (such as o i l and natural gas or c o a l ) , i t i s necessary to pro ject long-run energy pr ices fo l lowing the same approach as in the case of copper pr ices (see Pindyck[1976] and Leont ie f - 226 -[1977] fo r energy p r i ce p ro jec t i on s ) . There i s no d i f f i c u l t y in incorpor -a t ing transport costs and energy costs i f they are proport ional to output and machinery r e s p e c t i v e l y ; pr i ces p and g may be defined e i t he r net or gross of unit t ransport and energy cost r e spec t i ve l y . More elaborate s p e c i f i c a t i o n s are pos s ib le , but f o r s i m p l i c i t y and comparabi l i ty with the continuous time planning model, they w i l l not be considered in the 22 fo l lowing d i s c re te s p e c i f i c a t i o n of our planning model. 8.3 A Discrete Time Approximation to the Planning Model Since economic data come in d i s c re te form and d i g i t a l computer so lut ions require some kind of ' d i s c r e t i z a t i o n ' the planning model of chapter 3 must be approximated by a d i s c re te time optimal control or dynamic programming formulat ion. We have already s p e c i f i e d a l l the required funct iona l forms and provided a d i s c re te approximation to the object ive func t ion . There remains the tasks of taking e x p l i c i t account of the boundary cond i t i ons , convert ing the d i f f e r e n t i a l equations of the dynamic const ra int s into 23 d i f f e rence equations and der i v ing the d i sc re te - t ime opt ima l i ty condi t ions and costate equations. Note that we maintain the notat ion of chapter 3 inc lud ing the convention: g B P M , w E P^, m = m* and £ E a*. In a l l other respects the notat ion i s the same throughout the study (see Key Notation on page x i v ' ) . Note a l so that the symbol ' " ' over a parameter ind icates that the parameter has been e m p i r i c a l l y estimated ( a l te rna t i ve sets of values of the estimated parameters are found in chapter 7). Symbol ' ^ ' on the other hand, ind icates that i t s value i s assumed to be a given constant and s e n s i t i v i t y ana lys i s around th i s value i s necessary. - 227 -The complete d i s c re te optimal contro l model reads as fo l lows: Maximize the object ive func t i on , a. J ° = I (1 + ? ) _ t 4jr x l " ^ (8.1a) t=l 1-n with respect to the cont ro l s (x,v,q,w) and subject t o : ( i ) the boundary condi t ions (or i n i t i a l and terminal stocks) k-j = k-, , z^= z-j , r 1 = r 1 (8.8) k T+l = k T + T ZT+1 = ZT+1 ' r T+l = r T+l ^ 8 " 9 ^ and 23a ( i i ) the state equations (or dynamic cons t ra in t s ) k t + 1 = ( l - n-5 ) k t . + * ( k t , h t ) + ( p t - g t m t ) q t - x t - v t (8.10) z t + 1 = (1 - n - 6 ) z t + v t • - (8.11) r t + ] = (1 - n ) r t - q , (8.12) where <J>(kt,ht) = exp{6 0 + in ht + b^ Jin(k t/h t) - .5 b K N [ £ n ( k t / h t ) ] 2 + b T in T t > (8.13) h t = I - itqt (8.14) c. £t=w7[1 " g M + aML VK/Wt' " SZM £ n Z t - a R M an R t] (8.15) 228 -C t = e x p { a o + a M £ n ( 9 t / w t ) ' * 5 S M L ^ " ( g t / w t ^ ' + a Z M ^ Z t An(g t /w t ) * a R M £n R t J in(g t /w t ) + a z £n Z t + a R £n R t + .5 a z z U ' n Z t ) 2 + a Z R zn Z t £n R t + .5 a R R ( £ n R t ) + a T £n T t> ' t = [ a M " SML + aZM £ n Z t + aRM £ n R t ] Z t = V E t R t = V E t E t + 1 = (1 + n ) E t P t + 1 = (1 + e ) p t 8.16) 8.17) 8.18) 8.19) 8.20) 8.21) The necessary condit ions f o r maximizing the object ive funct ion (8.1a) subject to the boundary condi t ions (8.8) and (8.9) and the state equations (8.9) to (8.11) are der ived by assuming an i n t e r i o r so lu t ion and forming the discrete. Hamiltonian, H*: H t = —^-t -4-. x ^ + 3 t + 1 [ ( l - ? ( - 6 ) k t + 9 ( k t , h t ) (1 + p) I, - n + (P t - g t m t ) q t - x t - v t ] + Y t + 1 [ ( l - n - 6 ) z t + v t ] + ^ t + l C ( 1 ~ " ) r t " q t ] ( 8 - 2 2 ) Employing the d i s c re te - t ime maximum p r i n c i p l e (see Hal kin [1966]) we der i ve : - 229 -the opt ima l i ty condi t ions (or a l l o c a t i o n ru les ) x t = • [ ( H ? ) * 3 t + 1 ] - ^ (8.23) 3 t+ l = Yt+1 (8.24) *t+l / 8 t + 1 = P t - g tm t - • h t t (8.25) *h = (8.26) the costate equations (or zero p r o f i t condi t ions) Bt+1 r-i ^ ^ - i - l = [1 + 4>k - 6 - n] ' e t (8.27) Yt+1 = [1 + * n ^ z Q t + 9 t m z Q t - 3 - n ]_ 1 Y T (8.28) *t+l " [ 1 + ( P t - W V t Qt> " n ] *t (8.29) *k = * ( k t , h t ) . k [ b K " b K N ( k t ' h t ) ] (8.30) mw = £ t ( m t / c t ) + a M L ( c t / g w ) (8.31a) £w - £ t / c t - V w t " a M L ( c t / w t } (8.31b) m z = c z ( m t / c t ) + a Z M ( c t /w t Z t ) (8.32) C Z = c t j- [ a z + a Z M £n(g t /w t ) + § z z £n Z t + a Z R £n R t] (8.33) h = c z ( £ t / c t ) - a Z M ( c t / w t Z t ) (8.34) mR = c R ( m t / c t ) + a R M ( c t / g t R t ) _ (8.35) - 230 -c. C R = [ 1 " a Z + aRM m ^ t N t ] - + aRR m R t + aZR * n Z t ( 8 ' 3 6 ) h = " a R M ( c t " W t R t > ( 8 - 3 7 ) Note that subscr ipts t , and t+1 ind i ca te time per iods, while a l l other subscr ipts ind ica te p a r t i a l de r i va t i ve s of funct ions with respect to the var iab le ind icated in the subscr ip t . Note a l so that a l l p a r t i a l der i va t i ves re fe r to period t . The var iab le x i s a trend var iab le def ined d i f f e r e n t l y in a l t e r n a t i v e s p e c i f i c a t i o n s of the production technologies (see Chapter 7 and in p a r t i c u l a r Tables 7.2, 7.4 and 7.10). Summarizing, to f i n d the contro l sequence (x-j ,v-| ,q^ ,w^), ( x g s V g ^ * ^ ) ••• (x-r»vj»Qj>wj) that gives a s ta t ionary value fo r the performance J ° . ( i . e . , maximizes the sum of discounted u t i l i t i e s over the planning horizon) the planner must choose at each per iod : (a) the three state va r i ab l e s , k t , z t , and r^. (b) the three costate v a r i a b l e s , g^ -, y^, and ^> and, (c) the four contro l v a r i ab l e s , x^, v^, q t , w^. to s a t i s f y s imultaneously, ( i ) the boundary condi t ions (8.8) and (8.9) ( i i ) the three state equations (8.10), (8.11) and (8.12) ( i i i ) the four op t ima l i t y condi t ions (8.23) through (8.26) ( iv ) the three costate equations (8.27), (8.28) and (8.29). 24 8.4 So lut ion Algorithms Finding a set of values f o r a l l s t a te , costate and contro l var iab les that s a t i s f y s imultaneously a l l four condi t ions ( i ) through ( iv) above, i s not an easy task. A l l ex i s t i n g i t e r a t i v e procedures are based on not - 231 -s a t i s f y i ng some of the cond i t ions . A nominal so lu t ion (or nominal control h i s tory) i s chosen which s a t i s f i e s only a subset of these four cond i t i ons ; then through successive l i n e a r i z a t i o n the nominal so lu t ion i s modified so that i t comes c l o se r to s a t i s f y i n g the v io l a ted cond i t i ons . There are f i f t e e n poss ib le numerical methods that involve i t e r a t i v e procedures but so f a r only the three methods shown i n Table XXXI have been used extens ive ly : Table XXXI ITERATIVE PROCEDURES FOR NUMERICAL SOLUTION Condit ions S a t i s f i e d by Nominal So lut ion  Numerical State Costate Opt imal i ty Boundary Method Equations Equations Condit ions Condit ions Neighboring Extremal algorithms Yes Yes Yes Yes Gradient algorithms Yes Yes No No Quas i l i nea r i za t i on algorithms No No Yes Yes Source: Bryson and Ho [1975 : 213] The neighboring extremal algorithms work as fo l lows: - 232 -Step 1: Guess some i n i t i a l values for the costate var iab les ( f r ^ z ^ z ) • Use these values along with the i n i t i a l stocks (RpZ^,r^) to solve the opt ima l i ty condi t ions (8.23) through (8.26) fo r the contro l s (x-|,v-j ,q-j sw^). Use (i<i ...) and (x-| ...) to obtain {k^z^z^ from the state equations (8.10) - (8.12). Set t = 2. Step 2: Use ( R t , z t , r t ) and (eT»YT»*T) to obtain ( e t + 1 , v t + 1 » * t + 1 ) from the costate equations (8.27)-(8.29) and (x^v^q^w^. ) from the opt ima l i ty condi t ions (8.23)-(8.26). Step 3: Use ( k t » z t , r t ) and ( x t , v t , q t , w t ) to obtain ( k t + - j » z t + - | > r t + i ) f r o m the state equations (8.10)-(8.12). Set t = t+1. Step 4: Repeat steps 2 and 3 un t i l the f i n a l state var iab les (ky+-| > Z T +] > r-r+^) and the corresponding costate var iab les (3j +-| »Y t +-| ) are obtained. Then examine how fa r (kj +-| >zy+-| > rj+i) m ' ' s s t h e i r respect ive boundary (terminal) condit ions (k T +-j >ZT+-| ' ^ J + T ) given by (8.9) and use th i s information to modify the i n i t i a l (B£'"^2'^2^' Step 5: Repeat steps 1-4 un t i l the f i n a l values of the state var iab les 25 are 'c lose enough1 to the boundary condi t ions (terminal s tocks ) . The main d i f f i c u l t y with the neighboring; extremal algorithms i s the high s e n s i t i v i t y of the state var iab les at the terminal time to changes in the guesses of the i n i t i a l costate va r i ab le s . Transmission of information about appropriate changes in ( & 2 ^ z 3 ^ 2 ^ ^ r o m ^ n e terminal time to the i n i t i a l time i s subject to problems of numerical round-off e r ro r a r i s i n g from the ' ad jo in t nature' of the state and costate equations when l i n e a r i z i n g 26 about the optimal path. Since these two sets of equations are coupled together " i t i s not unusual f o r the numerical in tegrat ion with poorly - 233 -guessed i n i t i a l cond i t i ons , to produce ' w i l d ' t r a j e c t o r i e s in the state 27 space" g i v ing r i s e to s tate values that "exceed the numerical range of 28 the computer!" Thus the p rac t i c a l a p p l i c a b i l i t y of t h i s a lgorithm i s l im i ted to ( i ) so lv ing simple contro l problems with only one state var iab le and ( i i ) f i nd ing neighboring extremal so lut ions a f t e r one extremal so lu t ion i s obtained by some other a lgor i thm. The gradient algorithms were developed as a remedy to the ' i n i t i a l guess' problem of the neighboring extremal a lgor i thms. The gradient algorithms begin with nominal contro l h i s t o r i e s that s a t i s f y the state and costate equations and operate by i t e r a t i v e l y improving these contro l h i s t o r i e s to s a t i s f y the opt ima l i ty and boundary cond i t ions . The boundary condi t ions are approached by adding to the ob ject i ve funct ion (8.1a) quadrat ic penalty funct ions of the form: on deviat ions of the f i n a l values of the state var iab les (k-j-+-| » z j + - | »r-r+-|) from t h e i r target values (k T +-| ) 9 i v e n by the boundary condi t ions (8.8) and (8.9). There are a number of gradient a lgor i thms. In economic planning models the most extens ive ly used gradient algorithm goes as fo l lows: Step 1: Pick a nominal contro l h i s tory (x-| ,q-| ,v-| ,w-j) ( x g ^ ^ V g ^ ) , ( x T , q T , v T , w T ) . * a k ' k T + l " R T + l | 2 h a z | z T + 1 - i T + ] | 2 "s « - r l r T + l " F T + l ' 2 (8.38) (8.39) (8.40) - 234 -Step 2: Using t h i s h i s tory and the boundary ( i n i t i a l ) condi t ions (R-j ,z-j ) 29 from equation (8.8) ' i n teg ra te ' the state equations (8.10)-(8.12) forward in time to obtain a state h i s tory (R-j ), ( l ^ ^ , . ^ ) , •. •, C<T+-] »Zj+-| >rj+]) Step 3: Using ( k T + 1 , z T + 1 , r T + 1 ) and the penalty funct ions (8.38)-(8.40) evaluate the terminal costate var i ab les (&j +-| »YJ +-J ^y+ i ) a s fo l lows : 3 T + 1 = a k I k T + l ' W (8-41) Y T + 1 = a z l Z T + l ' Z T + l l <8- 4 2> * T + 1 = « r | r T + 1 - r T + 1 | (8.43) Step 4: " Integrate" the costate equations (8.27)-(8.29) backward in time to obtain a costate h i s to ry (e-j »Y-J »*-|)» ( 6 2 , Y 2 » * 2 ^ • • • ' ( eT+l 'YT+1 '^T+l ^ Step 5: Using the ca l cu la ted state and costate h i s t o r i e s c a l cu l a te fo r each period the Hamiltonian (8.22) and i t s gradient with respect to the c o n t r o l s , (3H /3x, 3H/3v, 3 H / 3 q , 3H/3w) , given by the opt ima l i t y condi t ions (8.23) to (8.26). Step 6: Make a one-dimensional search in the gradient (or modif ied 30 gradient) d i r e c t i o n un t i l the Hamiltonian i s maximized ( i . e . , i t s gradient becomes equal to ze ro ) . This involves ( i ) choosing a parameter p to . . . 31 minimize f (y ) = H[X - yVH'(X)] (8.44) where X H i s the current contro l h i s to ry vector i s the Hamiltonian i s the gradient of the Hamiltonian ind icates vector t ranspos i t i on - 235 -and ( i i ) c a l c u l a t i n g a new contro l h i s t o ry : Xnew ~ ^ x T v i ' q l ' w l ^' * * " ( x T ' v T ' q T ' w ' l : ^ f r o m ' t n e r e l a t i o n s h i p : Xnew = X " y V H ( X ) ( 8 > 4 5 ) Step 7: A f te r appropriate modi f icat ions in the parameters c ^ , a z , and a p of the penalty funct ions return to step 2 with the new control h i s to ry . The number of i t e r a t i o n s necessary fo r reaching an "optimal 1 so lut ion depends on the nominal control h i s to ry chosen in step 1 and the stopping c r i t e r i a s p e c i f i e d (e . g . , the admiss ible degree of divergence of the terminal s tate var iab les from t h e i r target va lues ) . F i r s t - o r d e r gradient methods are known to converge r ap id l y in the f i r s t few i t e r a t i o n s but show progress^-i v e l y d iminish ing performance as the optimum so lu t ion i s approached. Once a near optimum so lut ion i s obtained other methods such as the neighboring extremal methods and second-order gradient algorithms (which show exce l len t convergence near the optimum) may be used to come c lo ser to the optimum. Kendrick and Taylor [1968][1970][1971], however, have found that " f i r s t - o r d e r methods w i l l t y p i c a l l y determine contro l h i s t o r i e s 32 to three s i g n i f i c a n t d i g i t s in a score of i t e r a t i o n s " . They a lso found i t i s f ea s i b l e to solve models with f i v e or more state var iab les and nine or more contro l s and that "adding state and control var iab les in a c o n t r o l -theory model seems to increase computation time per i t e r a t i o n at a roughly 33 l i n e a r f a sh ion . " Since our model has only three state var iab les and four cont ro l s there should be no problem in obta in ing a numerical so lu t ion through the descr ibed gradient a lgor i thm or some quasi 1 inear i za t i on method. (The l a t t e r involves choosing nomina l - s ta te -var iab le h i s t o r i e s that s a t i s f y - 236 -the boundary cond i t i ons , computing the contro l vector from the opt ima l i ty 34 condit ions and modifying the nominal so lu t ion un t i l the ' l i n e a r i z e d ' oc state and costate equations are s a t i s f i e d ) . However, the required inputs of programming l o g i c and computation time are l i k e l y to be f a r greater in our case than in previous models of the same dimensions f o r a number of reasons. F i r s t l y , the production technologies and t h e i r pa r t i a l der i va t i ves appearing in the opt ima l i t y condi t ions and in the costate 36 equations are not simple funct ions (as in previous models ) but complex mu l t i f ac to r r e l a t i o n s h i p s , (8.13)-(8.17) and (8.30)- (8.37). These equations must be solved during each and every i t e r a t i o n . Secondly, because of the large number of exogenously s pec i f i ed parameters ( p , 5 , n , £ , n , e , x , T , k y + i » z y + i » a n d r j+ ] ) extensive s e n s i t i v i t y ana lys i s i s warranted. Once an ' opt ima l ' bas ic so lu t ion i s obtained these parameters should be var ied (one. at ..a time) around t h e i r basic values and the whole a lgor i thm should be repeated un t i l the corresponding 'mod i f ied ' so lut ions are obtained. Moreover, a l t e r n a t i v e funct iona l forms e s p e c i a l l y of the soc ia l welfare funct ion .should be attempted. Only i f these modified so lut ions in comparison to the basic so lu t ion ind ica te r e l a t i v e i n s e n s i t -i v i t y of the ' opt ima l ' t r a j e c t o r i e s to the assumed parameter values should the numerical re su l t s be taken se r i ou s l y . While a complete numerical so lu t ion requires a large s imulat ion pro ject which i s beyond the scope of the present study, the centra l planner (unl ike the independent researcher) need not be concerned with most of the aforementioned s e n s i t i v i t y ana l y s i s . In optimal growth models, as Sen [1967] pointed out, at l ea s t three p o l i t i c a l dec i s ions are invo lved: ( i ) the choice - 237 -of the soc ia l welfare funct ion : ( i i ) the length of the planning hor izon; and ( i i i ) the choice of the terminal stocks. Assuming that no separation of i d e n t i t y between p o l i t i c a l and planning au tho r i t i e s e x i s t s , the planner should have a f i r s t - h a n d knowledge of these three most important aspects of optimal development planning. Economics has l i t t l e to say on how these choices should be made, except that there appears " to be no reason not to make the planning horizon as long as po s s i b l e , subject to cost cons iderat ions and the 37 dec l i n i ng c r e d i b i l i t y of data f o reca s t s . " Economic ana l y s i s , however, may invest i gate the impl i ca t ions of a l t e r n a t i v e p o l i t i c a l dec i s ions in terms of the f e a s i b i l i t y of the object ives and the opt ima l i t y of the p o l i c i e s 38 d i rec ted towards the achievement of these ob jec t i ve s . Such could be the scope of a s imulat ion study based on the theore t i ca l ins i ghts and empir ical foundations that our research hopefu l ly has cont r ibuted. - 238 -FOOTNOTES TO CHAPTER 8 1. Note that we preserve the notat ion of chapter 3. See key notat ion of page x i v . 2. Furthermore, the numerical so lu t ion may be described in such deta i l that a set of annual aggregate national accounts cons i s tent with the optimal po l i c y may be constructed. 3. Add i t ive s e p a r a b i l i t y or independence of u t i l i t i e s over time, u(x-| , x 2 , ... ,Xj) = u-|(x-|) + U2U2) + . . . + U J ( X J ) , may be based on one of the fo l lowing i n t e r p r e t a t i o n s , provided by Fr i s ch [1957] and A l l en [1933-34], r e spec t i ve l y : ( i ) the u t i l i t y funct ion i s determined to a degree that allows fo r only increas ing l i n e a r transformations of u, which i s equiva lent to assuming card ina l u t i l i t y based on F r i s c h ' s " i n t e r l o c a l choice axioms"; or ( i i ) u t i l i t y i s o r d i n a l , a l lowing monotonical ly increas ing transformations of u, at leas t one of which can be wr i t ten in add i t i ve form. See l a s t paragraph of the present sect ion on the p o s s i b i l i t y of re lax ing the independence or add i t i ve s e p a r a b i l i t y assumption. 4. Chakravarty [1969 ;.: 27]. 5. Manne [1971 : 471]. 6. Margl in [1967 : 145-146] uses the fo l lowing u t i l i t y func t ion : u(x) = -bx~ a where b and a are pos i t i ve constants. 7. As we have seen in sect ion 4.1, i t i s d i f f i c u l t to obtain s u f f i c i e n t condi t ions on the parameters of a t rans log funct iona l form to ensure that a l l r e g u l a r i t y condit ions are g l oba l l y s a t i s f i e d , unless the r e s t r i c t i o n b.. = 0, i , j = l , . . . , n i s imposed i n which case (8.6) co l lapses to a Cobb-Douglas u t i l i t y func t ion . 8. Again there i s the d i f f i c u l t y of f i nd ing s u f f i c i e n t ( l e t alone necessary) conditions^on parameters a and b to ensure that these propert ies are g l oba l l y s a t i s f i e d . I t i s poss ib le to f i n d r e s t r i c t i o n s on a and b so that (8.6a) would s a t i s f y these propert ies f o r ce r t a i n ranges of x ( i . e . l o c a l l y ) . We did not, however, invest i gate these p o s s i b i l i t i e s ; we simply assume that they e x i s t . 9. Chakravarty [1969 : 340-841]. 10. The s p e c i f i c formulat ion used by Ryder and Heal [1973] may be wr i t ten in d i sc re te time as fo l lows: - 239 -CO J(x) I ( l + p ) _ t u[x(t) ,x(.t)] t=0 where x( t ) where e > 0 and X (T) i s the average leve l of per cap i t a l consumption in the community at time x . Then, the var iab le x( t ) may be regarded as the customary or expected leve l of consumption. 11. Chakravarty [1969 : 24]. 12. Central S t a t i s t i c a l O f f i ce [1970 : 23] National Accounts 1970, Lusaka, Zambia. 13. Mikesel l [1975 : 15]. 14. F i she r , Cootner, and Ba i l y [1972 : 598-599]. 14a. This i s a moving rather than a f i xed upper l i m i t . 15. See Ho te l l i ng [1931], Herfindahl [1961] and Scott [1967]. For an exce l l en t d i scuss ion of long-run mineral pr i ces and costs see Herf indahl [1959][1961]. 16. Ext rac t ion cost i s def ined here, as the long-run average cost of production (defined to include a l l stages from exp lorat ion to r e f i n i n g ) . As Herfindahl [1961 : 20] pointed out: "The cost of f i nd ing can be regarded as the same kind of cost as the cost of mining. That i s , i f our i n t e r e s t i s in the long sweep of events, expenditure on f i nd ing can be regarded as occuring approximately at the time of e x p l o i t a t i o n " . 16a. Herfindahl [1961:18] summarized the assumptions behind th i s r e su l t as fo l l ows : ( i ) a l l deposits o f the mineral are known and are of uniform q u a l i t y ; ( i i ) at some f in i ' te p r i c e , purchases of the metal drop to zero ; ( i i i ) the quant i t i e s demanded at various pr ices do not change over t ime; and ( i v ) per fec t competit ion p r e v a i l s , with many mines producing in any one year. 17. Constant returns to sca le are assumed. 18. See Pindyck[1976: Table 5 and Figure 4 ] . 19. Leont ie f [1977:5-6]. 20. See Leont ief [1977:. Table 75]. - 240 -21. Leont ie f [1977:73]. 22. For the same reasons a number of other points re levant to the Zambian case, such as the 49 percent fore ign share in mining as set s , which can be e a s i l y accommodated, have not been introduced. 23. In chapter 3 we employed the term ' op t ima l i t y cond i t i ons ' in a broad sense to inc lude a l l the necessary cond i t i ons , i . e . the four a l l o c a t i o n ru les and the three zero p r o f i t cond i t ions . Here, we use the term ' op t ima l i t y c o n d i t i o n s ' , in a narrow sense to r e f e r only to ~ the a l l o c a t i o n ru le s . The term 'costate equations ' i s used to re fe r to the zero p r o f i t cond i t ions . 23a. Note that for s i m p l i c i t y we assume 6-j = S 2 = 6 while the term ' s ta te equations ' re fe r s to the 'dynamic c o n s t r a i n t s ' . The so lut ion algorithms to be discussed in sect ion 8.4 can best be described using the new terminology. 24. This sect ion draws extens ive ly on Bryson and Ho [1975:212-245] and Kendrick and Tay lor [1968][1970][1971]. 25. Some convergence c r i t e r i a must be e x p l i c i t l y s p e c i f i e d . For instance, we may require that the d i f fe rence between f i n a l and terminal (or target) stocks be less than one percent of the target for at l eas t f i v e i t e r a t i o n s . 26. The fundamental so lu t ions of the two sets of equations may diverge over time by many orders of magnitude (see Bryson and Ho [1975:215]). 27,28. Bryson and Ho [1975:215]. 29. Since our model i s in d i s c re te time form we are a c tua l l y summing rather than i n teg ra t i ng . 30. The modi f icat ions aim at u t i l i z i n g information embodied in curvature propert ies of the Hamiltonian with respect to the con t ro l s . Kendrick and Taylor [1971] employed a 'conjugate gradient ' procedure which is based on so lv ing a sequence of one dimensional maximization problems. At each step, th i s procedure changes the contro l vector "according to a'weighted average of t h i s step gradient and l a s t ' s step d i r e c t i o n of movement" (Kendrick and Taylor [1971:26]). For a de ta i l ed account of the conjugate gradient methods see Powell [1964] and F le tcher and Reeves [1964]. 31. Equation (.8.44) i s minimized by s e t t i n g , df(y) dy vH(x)-[vH(x-yVH(x))] - 241 -equal to zero. For a s impler but more crude procedure for obtaining the value of y that minimizes (8.44) see Kendrick and Taylor [1971:26]. 32. Kendrick and Taylor [1970:459]. 33. Kendrick and Tay lor [1968:232]. 34. L inear i zed about the norminal s o l u t i on . 35. For d e t a i l s on the quasi l i n e a r i z a t i o n algorithms see Bryson and Ho [1975:234-236]. 36. For instance Kendrick and Tay lor [1971] and Vajed [1975] use two-input Cobb-Douglas production funct ions . 36a. X i s a scale parameter on output (see Appendix B) here assumed equal to 1, s ince constant return to var iab le inputs are assumed. This assumption may be relaxed as ind ica ted in Appendix B. 37. Kendrick and Tay lor [1970:461]. 38. An attempt to invest igate the f e a s i b i l i t y and degree of opt ima l i ty of Zambian resource and development planning in the context of an intermediate-run vers ion (1970-81) of the Model was abandoned when the pub l i ca t ion of the Th i rd National Development Plan (scheduled fo r 1976) was postponed. The Zambian M in i s ter of Planning was a l so unable to provide us with the requested data on target growth rates and terminal stocks. CHAPTER 9 SUMMARY AND CONCLUDING REMARKS The purpose of the study was twofold: f i r s t , to develop a model o f economic development through optimal mineral e x p l o i t a t i o n ; and second, to elaborate on those aspects of the model that have not s a t i s f a c t o r i l y been dea l t with in the l i t e r a t u r e , s p e c i f i c a l l y the formulation and est imation of the production technologies and the measurement of the stocks of natural and physical c a p i t a l . In th i s chapter we summarize and evaluate the main re=;"t^ s u i t s , discuss some l i m i t a t i o n s , and suggest ways in which the study might be extended. The theore t i ca l part of the thes i s models the optimal exp lo i t a t i on of a non-renewable heterogeneous resource in the context o f a developing economy. Opt imal i ty was defined in terms of r a i s i n g the current standards of l i v i n g without unduly depr iv ing future generations of the resource bene-f i t s . This involved not only a l l o c a t i n g the resource stock between the present and the f u tu re , in an optimal way; i t a lso required optimal a l l o -cat ion of the mineral revenues between current consumption and investment in reproducib le c a p i t a l . The l a t t e r took two forms: ( i ) investment in mining s t ructures to maintain mineral production in the face of de te r i o ra t ing resource q u a l i t y ; and ( i i ) investment in non-mining cap i t a l to create an i ndu s t r i a l base as an a l t e r n a t i v e to the deplet ing natural resource. The specia l features of the constructed intertemporal model are: ( i ) the d i s -t i n c t i o n between mining and non-mining production technolog ies ; and ( i i ) the d i s t i n c t i o n , with in the mining technology, between f i xed and va r i ab le i n -puts, on the one hand, and between domestic and imported inputs , on the other. -2.43-Despite the complex nature o f the problem we were able to der ive some e