The Efficiency of the Coal Industry: An Application of Linear Programming [Reprint 2014 ed.] 9780674493179, 9780674493162

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HARVARD ECONOMIC

STUDIES

Volume C H I

Awarded the David A. Wells Prize for the year 1955-56 and published from the income of the David A. Wells Fund. This prize is offered annually, in a competition open to seniors of Harvard College and graduates of any department of Harvard University of not more than three years standing, for the best essay in certain specific fields of economics. The studies in this series are published by the Department of Economics of Harvard University. The department does not assume responsibility for the views expressed.

THE EFFICIENCY OF THE COAL INDUSTRY An Application of Linear Programming

By

JAMES M. HENDERSON

HARVARD UNIVERSITY PRESS Cambridge, Massachusetts 1958

© C o p y r i g h t 1958, b y the PRESIDENT AND FELLOWS OF HARVARD COLLEGE

Distributed in Great Britain by Oxford University Press, London

Printed in G r e a t Britain.

To my father

Preface THIS study is a result of the author's interest in the economic problems of natural resources and his desire to bring the theoretical and empirical sides of economics closer together. Theory and quantitative analysis are developed side by side, and an attempt has been made to give numerical content to the basic theoretical postulates and results. Linear programming provides the framework within which this task has been carried out. The basic techniques utilized for the coal industry are described in detail with the hope that they will prove useful for others who desire to pursue similar studies. The author owes his greatest debt of gratitude to Professor Wassily W. Leontief. He provided the major intellectual stimulus for the present study and was the source of many valuable suggestions as it passed through its several stages. The author has also benefited from discussions with Professors John S. Chipman and Robert Dorfman concerning the use of linear programming methods. The research underlying this volume was begun in 1953 while the author was the holder of an Earhart Foundation Fellowship. It was completed in 1955 as a part of the research program of the Harvard Economic Research Project while the author was a member of its Senior Research Staff. The author is grateful to the members of the staff of the H E R P for both specific suggestions and many fruitful discussions. Mrs. Elizabeth W. Gilboy read the entire manuscript and provided aid for both editorial and substantive problems. Mr. Sven Dano also read the entire manuscript. Professors John Fei (now at Antioch College) and Leon N. Moses were prominent among those with whom the author discussed his problems. Mrs. Barbara King helped prepare the index. vii

Preface The author is obliged to Professor Seymour E. Harris, managing editor of The Review of Economics and Statistics, for permission to quote previously published materials. Cambridge, Mass. November, 1957

J. M. H.

viii

Contents I. A M E T H O D O F A N A L Y S I S The Competitive Norm - Deviations from the Competitive Norm - Linear Programming - Efficient and Competitive Norms for the Coal Industry II. A S H O R T - R U N M O D E L F O R T H E COAL INDUSTRY Classification of Deposits and Consuming Areas Demand Requirements - Capacity Restrictions - Extraction Costs - Transport Costs - The Allocation Problem - The Programming Format - The Dual Problem A Method of Solution III. APPLICATIONS OF T H E MODEL Output Index - Classification of Deposits - The Data Tests of the Model IV. T H E O P T I M U M

SOLUTIONS

Stability of the Short-Run Solutions - Comparative Statics - Errors of Observation - Solutions of the Dual System V. M E A S U R E S O F E F F I C I E N C Y A N D D I S T R I BUTION Measures of Efficiency - The Distribution of the Costs of Inefficiency VI. S E C U L A R I N E F F I C I E N C Y The Conformity of Surface Outputs - Regional Impacts of Secular Changes - The Organization of the Coal Industry - Possible Capacity Reductions - Methods of Control

ix

Contents VII. C Y C L I C A L I N E F F I C I E N C Y

106

Output Declines: 1947 to 1949 - Union Restrictions Conclusions VIII. SUMMARY

113 APPENDICES

A. D A T A : S O U R C E S A N D R E C T I F I C A T I O N

119

Demands - Capacities - Extraction Costs - Transport Costs - Conversion to Heating Value Units B. A S H O R T - R U N M O D E L F O R A G R I C U L T U R E Classification of Land - Demand Requirements Capacity Restrictions - Cultivation Costs - Transport Costs - The Allocation Problem - Programming Format Dual Program

133

A SELECTED BIBLIOGRAPHY

139

INDEX

145

x

Tables 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Demands and Capacities for the Hypothetical System Relations of the Hypothetical System Unit Costs for the Hypothetical System Relevant Unit Costs and Reduced Capacities and Demands after Selection of Three Deliveries for the Initial Solution Initial Basic Feasible Solution Opportunity Costs for the Initial Basic Feasible Solution Second Basic Feasible Solution Opportunity Costs for the Second Basic Feasible Solution Third Basic Feasible Solution Opportunity Costs for the Third Basic Feasible Solution Coal Districts and Relative Coal Outputs in 1951 Demands for Coal: 1947, 1949, and 1951 Capacities by Districts and Methods of Mining: 1947, 1949, and 1951 Unit Extraction Costs by Districts and Methods of Mining: 1947, 1949, and 1951 Unit Costs of Extraction and Transportation: 1947, 1949, and 1951 Variance Ratios for Linear and Homogeneous Functions Algebraic Signs of the Deviations for Underground Mining: 1946-1952 Percentages of Underground Mined Coal Mechanically Loaded, by Districts: 1946-1952 Minimum Cost Solution for 1947 Minimum Cost Solution for 1949 Minimum Cost Solution for 1951 xi

29 31 31

33 34 36 38 38 39 39 44 45 47 49 51 57 58 59 62 64 66

Tables 22. Changes in the Minimum Cost Solution for 1947 Resulting from the Introduction of a Delivery from the Underground Deposit in District 1 to District 5 68 23. Effects of a 50-Percent Increase in the Demand of District 9 upon the Efficient Solution for 1951 71 24. Included Deliveries with Narrow Cost Differentials: 1947, 1949, and 1951 76 25. Competitive Unit Royalties: 1947, 1949, and 1951 78 26. Competitive Delivered Prices: 1947, 1949, and 1951 78 27. Actual and Efficient Outputs: 1947, 1949, and 1951 82 28. Deviations of Actual from Efficient Outputs: 1947, 1949, and 1951 84 29. Actual and Efficient Costs of Extraction and Transportation: 1947, 1949, and 1951 85 30. Actual and Competitive f.o.b. Mine Prices: 1947, 1949, and 1951 88 31. Average Number of Days Active, Surface and Underground Deposits: 1947-1953 97 32. A Possible Reduction of the 1951 Capacity of the Coal Industry by 200,000 Units 101 33. Number of Bituminous Coal Mines, and Net Increase {Decrease) in Number and Total Capacity: 1916— 1952 104 34. Actual and Efficient Output Changes between 1947 and 1949 107 35. Efficient Outputs and Minimum Deviations for 1949 Adjusted for Union-Imposed Capacity Restrictions 110 A-l. A-2. A-3. A-4. A-5.

Estimated Demands: 1947, 1949, and 1951 Estimated Capacities: 1947, 1949, and 1951 Estimated Unit Extraction Costs : 1947, 1949, and 1951 Estimated Unit Transport Costs : 1947, 1949, and 1951 Heating Value Conversion Factors: 1947, 1949, and 1951

121 123 125 130 132

Charts 1. Capacity and Output of the Coal Industry: 1904-1953 2. Percentage of Capacity Utilized: 1904-1953 xii

94 95

The Efficiency of the Coal Industry: An Application of Linear Programming

CHAPTER I

A Method of Analysis MODELS of perfect competition dominated economic theory throughout the neoclassical period of the late nineteenth and early twentieth centuries, and have persisted in one form or another to the present day. Many of the major contributions to theoretical economics in recent years are based upon the assumptions of perfect competition.1 The current prevalence of perfect competition is witnessed by the fact that its postulates and results are familiar to every student of elementary economics. A perfectly competitive industry contains a large number of firms, none of which is able to exert a significant influence upon the market price of its products. The perfectly competitive firm accepts its production function and the prices of its inputs, as well as the price of its output, as data. Perfect knowledge and factor mobility are assumed. Each firm selects an output level which maximizes profit under these conditions. An industry supply curve is constructed by summing the marginal cost curves of the individual firms, and product price is determined by its intersection with an aggregate demand curve. Competition among existing firms, together with the free entry of new firms and the free exit of existing firms, carries production to a level at which pure profit equals zero throughout the industry. Production functions are generally assumed to possess continuous partial derivatives of at least the second order, but are seldom given specific shapes. Firms are allowed to vary the quantities and proportions of the factors that they employ until an equilibrium situation is reached in which the price of each factor equals the value of its marginal physical product. 1 Two outstanding examples are J. R. Hicks, Value and Capital (2nd ed.; Oxford, 1946); and Paul A. Samuelson, Foundations of Economic Analysis (Cambridge, Mass., 1947).

1

The Efficiency of the Coal Industry The marginal productivity theory of distribution follows from competitive behavior in the productive sphere. Price equals marginal cost, which in equilibrium equals minimum average cost. Factors of like quality are assumed to receive uniform compensation throughout the economic system. Factor prices measure the values of the factors' contributions at the margin of production, that is, the values of their marginal physical products. Generally the results of perfect competition are assumed to be reached automatically in a free market. The prices and outputs that are determined in a perfectly competitive system are definite quantities, but they are hardly ever numerically determined. The quantitative nature of this analysis is limited to general marginal equalities. Perfect competition can be described by either its necessary conditions or its results. Some of the more important necessary conditions are (1) large numbers of buyers and sellers, (2) the acceptance of product and input prices as parameters, (3) the absence of selling costs, (4) freedom of entry and exit, (5) perfect knowledge, and (6) perfect factor mobility. Perfect competition is an "ideal type" in that its conditions are never completely fulfilled in any actual market situation. Its postulates can only serve as an approximation of an actual situation. A number of economists judge the nature of such approximations on the basis of the closeness with which the necessary conditions of perfect competition are approximated in actual situations. Another group of economists judge the descriptive powers of perfect competition on the basis of the closeness with which actual results approximate those of perfect competition. Some of the more important results of perfect competition are (1) the equality of prices and marginal costs, (2) the equality of factor prices and values of marginal physical products, and (3) the existence of zero pure profit levels. These results, as well as the necessary conditions, are never completely realized in actual situations. However, one could make a judgment on the basis of results which is quite different from a judgment made on the basis of necessary conditions. Economists interested in the descriptive powers of economic theory have revised the theory of perfect competition in an attempt 2

A Method of Analysis to bring economic theory into closer accord with actual situations, in other words, to explain price and output determination under conditions of nonperfect competition. The theory of monopoly is perhaps older than the theory of perfect competition. Chamberlin's Theory of Monopolistic Competition 2 and Mrs. Robinson's Economics of Imperfect Competition 3 are attempts to develop general theories of nonperfect competition. Numerous theories of duopoly and oligopoly are also in existence. Unfortunately, there appear to be as many theories of nonperfect competition as there are special situations. Chamberlin's work is the most general. THE COMPETITIVE NORM

Aside from their somewhat dubious value as approximations of actual situations, the results of perfect competition serve as implicit, if not explicit, welfare norms for many economists. The allocation of factors which results from perfect competition is statically efficient in the sense that the marginal contribution by each factor equals its price in every employment, and all outputs are produced at minimum average cost. No other factor allocation is possible in which the welfare of one individual can be increased without a reduction in the welfare of at least one other individual. Situations of nonperfect competition are inefficient within this context. Lemer, perhaps, has made the most extreme use of the norm of perfect competition as a welfare ideal.4 He proposes to force the static efficiency of perfect competition upon the economy through a set of marginal rules to be enforced by a central authority. However, he is not satisfied with distribution under perfect competition, and proposes a divorce of the distributive norm from the efficient. Others, less extreme than Lerner, accept the welfare ideal of perfect competition, and look upon actual situations in terms of their deviations from this norm. The Theory of Monopolistic 2

Edward Hastings Chamberlin, Theory of Monopolistic Competition (7th ed.; Cambridge, Mass., 1956). 3 Joan Robinson, Economics of Imperfect Competition (London, 1933). 4 A. P. Lerner, Economics of Control (New York, 1946).

2

3

The Efficiency of the Coal Industry Competition has been considered by many economists as a major contribution to welfare economics because it leads to the conclusion that firms operate to the left of the minimum points of their average cost curves. Mrs. Robinson reasoned directly along welfare lines when she titled her work The Economics of Imperfect Competition. A major part of her attention is devoted to "exploitation" (whereby factors receive less than the values of their marginal physical products) under imperfect competition. DEVIATIONS FROM THE COMPETITIVE NORM

No industry conforms exactly to all of the conditions and results of perfect competition, and the question to be answered in the analysis of a particular industry is not if it satisfies these conditions and results but rather how closely it satisfies them. The extent of competition in individual industries has been the subject of a large number of studies. A common procedure is to examine a number of the necessary conditions, and then judge the competitive structure of an industry on the basis of qualitative comparisons of these with actual conditions.5 These analyses provide valuable narratives which describe the nature of competition within particular industries, but tell little about results. The extent to which actual results deviate from those of perfect competition may not be directly related to the number of noncompetitive practices that can be listed, since the effects of one practice might offset those of another, and the same practice can cause quite different results under different circumstances. A few economists have attempted to measure some aspect of the deviation of an actual market situation from the perfectly 5

A study of the bituminous coal industry (the industry in which our attention shall soon be centered) which follows this general procedure has been made by Jacob Schmooker, "The Bituminous Coal Industry," in Walter Adams (ed.), The Structure of American Industry (rev. ed.; New York, 1954), pp. 76-113. Schmooker describes the coal industry as imperfectly competitive. Another author, whose major interest is in labor relations, describes it as "workably competitive," Morton S. Baratz, The Union and the Coal Industry (New Haven, 1955), p. 27. Still another author has noticed tendencies toward both monopoly and competition, Glen L. Parker, The Coal Industry: A Study in Social Control (Washington, 1940), chap. i.

4

A Method of Analysis competitive norm. These studies deal either with a necessary condition, or a result. Some measures are operational in the sense that they can be applied with the use of available information, and others are not. Measures of industrial concentration deal with the necessary condition of a large number of sellers, and are operational. These analyses usually attempt to measure the concentration of ownership within an industry with regard either to sales or assets, but they "... explore concentration in the structural sense only, without regard to its behavior consequences." 6 If one is interested in results rather than institutional framework, measures of industrial concentration are of little assistance. The so-called "measures of monopoly power" represent attempts to measure the deviation of some aspect of the results of an actual situation from that which would prevail under perfect competition. Monopoly in this sense means a deviation from perfect competition rather than the existence of a single seller. Lerner's measure was the first ^nd perhaps best known of these. 7 Marginal cost pricing is the essential element of perfect competition in Lerner's estimation, and his measure of monopoly power is the difference between price and marginal cost (divided by price in order to obtain a pure number). Lerner is interested in the "... divergence of the system from the social optimum that is reached in perfect competition." 8 The operationality of his measure is rather questionable.9 Bain revised Lerner's measure to allow empirical estimation by using the difference between the rate of profit and the normal interest rate as a measure of " monopoly power." 10 Theoretically, this is the difference between price and average cost. A number 6

M. A. Adelman, "The Measurement of Industrial Concentration," Review of Economics and Statistics, XXXIII (November 1951), p. 269. 7 A. P. Lerner, "The Concept of Monopoly and the Measurement of Monopoly Power," Review of Economic Studies, I (June 1934), pp. 157-175. 8 Lerner, "The Concept of Monopoly," p. 168. 9 The only attempt to apply a measure similar to Lerner's was made by John T . Dunlop, "Price Flexibility and the 'Degree of Monopoly,' " Quarterly Journal of Economics, LIII (August 1939), pp. 522-534. Dunlop dealt with changes in the gap between price and average prime cost. 10 Joe S. Bain, "The Profit Rate as a Measure of Monopoly Power," Quarterly Journal of Economics, LV (February 1941), pp. 271-293. S

The Efficiency of the Coal Industry of other measures—mostly nonoperational—based upon a variety of factors have been proposed. 11 Many other instances in which economists have endeavored to judge actual market situations in the light of a perfectly competitive norm could be cited, but the present list is long enough to indicate the importance of these norms in economic analysis. LINEAR PROGRAMMING

The present monograph is primarily devoted to a study of the deviations of the actual results of the bituminous coal industry from perfectly competitive norms. However, our method of analysis differs from those discussed above. A short-run competitive model for the coal industry is established within the framework of a linear programming problem. The model is implemented with actual data for three postwar years, and the normative (perfectly competitive) values of its variables for each year are numerically determined. The normative values of the variables are compared with their corresponding actual values in order to obtain numerical measures of the deviation of the coal industry from its competitive norms. The recent development of linear programming has made possible a new type of economic analysis.12 Its methods allow the determination of numerical solutions for theoretical problems involving economic choice. Linear programming provides a general mathematical model which is applicable to problems that can be expressed as the maximization (or minimization) of a linear form subject to a system of linear inequalities. Many 11 K. W. Rothschild, "The Degree of Monopoly," Economica, IX (February 1942), pp. 24-39; Theodore Morgan, "A Measure of Monopoly in Selling," Quarterly Journal of Economics, L X (May 1946), pp. 461-463; and A. G. Papandreaou, "Market Structure and Monopoly Power," American Economic Review, X X X I X (September 1949), pp. 883-897. 12 There is an extensive linear programming literature. The basic reference, referred to henceforth as Activity Analysis, is Tjalling C. Koopmans (ed.), Activity Analysis of Production and Allocation (New York, 1951). Other general works are Robert Dorfman, Application of Linear Programming to the Theory of the Firm (Berkeley, 1951); and A. Charnes, W. W. Cooper, and A. Henderson, An Introduction to Linear Programming (New York, 1953).

6

A Method of Analysis economic problems can be placed within this general format, and linear programming may be utilized for the analysis of a plant, a firm, an industry, a national economy, or the world economy. The significance of linear programming for economics is similar to that of the maximization principle of the calculus. Both provide general methods that are widely applicable. The model for the coal industry is an application of a special type of linear programming problem—the so-called transportation problem formulated by Koopmans and others. 13 Special methods have been developed which allow this type of problem to be solved with ease. 14 Even this special type of problem is susceptible to considerable modification for specific applications. The short-run model for the coal industry treats the extraction as well as the transportation of coal, and expresses the availability limitations for coal as inequalities rather than equalities as is customary for transportation-type problems. The model for the coal industry contains two separate, but interrelated, programming problems. The "short-run" with reference to the coal industry is a period of time during which the extraction functions for coal deposits, and the prices of the inputs used for the extraction of coal remain unchanged. The data of the model are spatially distributed demands for coal, the capacities of spatially distributed deposits of coal, and the unit costs of deliveries from the deposits to the demand locations. The levels of the deliveries are the variables of the first of the programming problems, and are selected to minimize the costs of meeting the demands subject to the capacity restrictions. The variables of the second problem, the dual system in the linear programming parlance, are the delivered prices of coal at the demand locations and the unit royalties earned by the various deposits. The values of these variables are selected to maximize total revenue net of royalty payments. The optimum solutions 13 Frank L. Hitchcock, "The Distribution of a Product from Several Sources to Numerous Localities," Journal of Mathematics and Physics, Massachusetts Institute of Technology, X X (1941), pp. 224-230; Tjalling C. Koopmans, "Optimum Utilization of the Transportation System," Econometrica, XVII (July 1949, supplement), pp. 136-145; and Tjalling C. Koopmans and Stanley Reiter, " A Model of Transportation," Activity Analysis, pp. 222-259. 11 George B. Dantzig, "Application of the Simplex Method to a Transportation Problem," Activity Analysis, pp. 359-373.

7

The Efficiency of the Coal Industry for both probLems are demonstrated to provide a complete description of a perfectly competitive short-run equilibrium for the coal industry. 15 The general nature of the model perhaps can be most easily illustrated with the aid of a simple hypothetical example. Imagine a closed economy with two coal deposits and three coal-consuming areas. Each consuming area has a fixed demand for coal, and each deposit has a fixed short-run capacity. Assume that each of the deposits has a unit extraction cost which is invariant with respect to the quantity of coal extracted from the deposit during a short-run period. These data of the model can be written in tabular form : Demands

Extraction

Area

Amount

Deposit

Capacity

Unit cost (dollars)

1 2 3

100 200 50

1 2

125 400

2.00 2.75

Further, assume that the following schedule of deposit-to-area transport costs is given by an outside regulatory agency : Unit transport costs (dollars) to area From deposit

1

1 2

1.00 2.00

2

3 1.50 1.00

2.00 1.75

The total unit cost of a delivery is the sum of its unit extraction and unit transport costs. A separate total unit cost exists for each of the six possible deliveries from the two deposits to the three consuming areas. A total cost schedule can be constructed by adding the appropriate extraction and transport costs: " See below, pp. 25-28.

8

A Method of Analysis Total unit costs (dollars) to area— From deposit

1

1 2

3.00 4.75

2

3 3.50 3.75

4.00 4.50

A feasible program is any set of deliveries which satisfies the demands and does not violate the capacity restrictions of the various deposits. An examination of the data given above reveals that there is a large number of such programs. An optimum program, or solution, is one which minimizes the total costs of extraction and transportation which are computed by multiplying each delivery level by its total unit cost and summing the products. This example can be solved in many different ways.16 The minimum cost solution is as follows: Deliveries From deposit 1 1 2 2 Unused capacity 2

To area 1 3 2 3

Level 100 25 200 25 175

Minimum total cost is 1,262.50 dollars. All three areas can be supplied at a lower cost by the first deposit than by the second. However, in the optimum solution the first area is supplied by the first deposit, the second area by the second, and the third area by both the first and second. An individual area is not necessarily supplied by the deposit which is least costly if that area is considered in isolation from the others. If the first deposit were to supply all three areas, it would have to deliver 350 units of coal, but its capacity is only 125 units. 16 If the reader is unable to derive the solution, he must accept it on faith for the present. The model is formally developed, and a general method for its solution is described in the next chapter.

9

The Efficiency of the Coal Industry The limited capacity of this deposit is allocated in the most efficient way. The cost differentials between deliveries from the two deposits to each of the three areas can be considered as the opportunity costs of deliveries from the first deposit. These are 1.75 dollars for the first area, 0.25 for the second, and 0.50 for the third. The 125 units of coal available at the first deposit are distributed 100 units to the first area where the opportunity cost is highest, and 25 units to the third area where the opportunity cost ranks second. The marginal cost of coal for a consuming area is defined as the minimum increase (maximum decrease) of total cost which would result from a unit increase (decrease) of the area's demand. 17 Marginal costs are 3.50 dollars for the first area, 3.75 for the second, and 4.50 for the third. The marginal costs for those areas (second and third) served by the second deposit, which has unused capacity in the optimum solution, equal the costs of deliveries from the second deposit. If the demand of one of these districts were increased (decreased) by one unit, the delivery from the second deposit would be increased (decreased) by one unit and its unused capacity decreased (increased) by one unit. The marginal cost for the first area can be obtained by computing a new minimum cost solution in which the demand of the first area is one unit more (less) than in the original problem. The results of such a computation can be seen rather easily for the present example. If the demand of the first area is increased (decreased) by one unit, the delivery from the first deposit to the first area is increased (decreased) by one unit, the delivery from the first deposit to the third area is decreased (increased) by one unit, the delivery from the second deposit to the third area increased (decreased) by one unit, and finally the unused capacity of the second deposit is decreased (increased) by one unit. The marginal cost is the sum of the unit costs of the deliveries that are increased less the unit cost of the delivery that is decreased: 3.00-4.00 + 4.50 = 3.50. The delivered price of coal in an area equals its marginal cost 17

In some situations the marginal cost of a unit increase differs from that of a unit decrease, but these situations are not common, and are not considered here.

10

A Method of Analysis for that area. The differential royalty earned by a deposit equals the delivered price of coal in an area to which that deposit delivers less the total unit cost of its delivery. In the present example, the first deposit earns a unit royalty of $0.50 ($3.50 — 3.00 and $4.50 — 4.00), and the second earns a zero unit royalty ($3.75 — 3.75 and $4.50 — 4.50). Deposits which are fully utilized in the optimum solution earn positive royalties, and those with unused capacities earn zero royalties.

EFFICIENT AND COMPETITIVE NORMS FOR THE COAL INDUSTRY

Optimum solutions for the two programming problems contained in the model provide numerical descriptions of the results of efficiency and perfectly competitive pricing. A solution for the delivery problem provides an efficient norm. Efficiency in this sense merely means that the prescribed demands cannot be satisfied at a lower total cost. The solution of the price, or dual, program contains those delivered prices and unit royalties which would result if the conditions and results of perfectly competitive pricing were realized. Previous competitive norms have been based upon the assumption that all sellers are faced with perfectly elastic demand curves. An individual seller can sell his entire output at the prevailing price; he cannot influence price appreciably by increasing or decreasing his output. Market demand curves, however, are assumed to slope downward. The market demand curves of the present model are perfectly inelastic, i.e., the level of demand in each area is invariant with respect to price. Prices do not reach infinite levels because of the competition between sellers and the desire of consumers to obtain their requirements at the lowest possible prices. The assumption of invariant demand levels is not as restrictive as might first appear. There are indications that, since the demand for coal is largely a derived demand, it may be quite inelastic for a rather wide price range during a short-run period. Furthermore, the price elasticity of the demand for coal is not of importance for our applications of the model to past periods in which demands are accepted as data. No attempt is made to 11

The Efficiency of the Coal Industry determine future conditions, and the question of the determination of demand levels is side-stepped. The solutions of the delivery program provide delivery levels which would have satisfied historical demands at minimum cost. Efficiency in the sense of minimum total cost is somewhat more restrictive than the traditional concept of competitive efficiency. The proportions in which factors are combined are assumed fixed in the short run. This assumption may not be more unrealistic than the traditional assumption of continuous factor adjustments at the margin. The model is a partial equilibrium analysis, and factor prices are also assumed fixed in the short run. Furthermore, the average extraction cost for each deposit is assumed constant for any attainable level of output. The model is empirically implemented with factual information for the demands, capacities, and unit costs for three postwar years, and numerical values are computed for the delivery levels, prices, and royalties for each of these years. The applications of the model contain twenty-two deposits and fourteen consuming areas. The observed values of the variables are compared with their normative values in order to determine how closely the coal industry approaches the norms of efficiency and competitive pricing established by the model. The level of efficiency of the coal industry is measured in terms of both output and cost conformity. Efficient output levels for the various deposits are aggregated from the efficient delivery levels, and compared with corresponding actual output levels. The total costs of the efficient and actual delivery levels are compared, and the costs of inefficiency are computed. These costs are separated into a cyclical component which varies with the level of total demand, and a secular component which persists from year to year. The two components are analyzed in detail with the aid of additional information. The prices and royalties determined in the model are analyzed together with available information covering their actual values in order to obtain an indication of the distribution of the costs of inefficiency among the consumers of coal, the mine operators, and the deposit owners. The coal industry is judged by its deviations from the efficient results of perfect competition. Necessary conditions are used as aids in the explanation of the divergency of results.

12

CHAPTER

II

A Short-Run Model for the Coal Industry OUR model for the coal industry deals with allocation in a shortrun period. The major distinction between a short-run and a long-run analysis lies in the number of variables which must be explained. Many slow-moving variables which can be assumed to remain constant for a short period of time may show considerable variation over a longer period. The capacities of the various coal deposits and their extraction functions are examples of such variables. These comprise a portion of the data for the present short-run model, but their changes over a long period are quite significant though largely unpredictable. Inferences covering long-run movements can often be drawn from a series of short-run solutions. This is particularly true in the present case because of the remarkable degree of stability shown by the short-run solutions for the coal industry. 1 Coal is a natural resource, and is distinguished from produced outputs by the conditions of its supply. Coal deposits are required as inputs to obtain an output of coal, but they are quite specific inputs in the sense that they can be used only for the extraction of coal. Unlike most other inputs, coal deposits are not allocated among alternative uses. The only decisions with regard to a coal deposit are whether it should be exploited, and if so, the extent of its exploitation. Coal in the ground is a free gift of nature, but is of no use to man until it has been extracted from its natural situation. The extraction of coal from a deposit can be described by the extraction function for that deposit. Coal deposits are frequently described only in terms of their recoverable reserves, but such descriptions by themselves are insufficient for economic analysis. Extraction 1

See below, pp. 69-70. 13

The Efficiency of the Coal Industry functions that can be transformed into cost functions must be specified before economic analysis can proceed and economic decisions be made. Coal is an exhaustible resource. Its stocks necessarily decrease with the process of extraction. Once coal is mined and used it is gone forever. Economists have formulated optimal time-use patterns for exhaustible resources.2 Such analyses are necessarily long-run. The exhaustibility of coal, however, is of little or no importance for a short-run analysis. The reserves of coal in the United States are so large that exhaustibility is of questionable importance even for a long-run analysis. Recoverable reserves of coal in 1953 were estimated to be 949,870 million tons, or more than two thousand times the rate of output for 1953. The complete exhaustibility of coal need not be considered. Exhaustibility becomes of importance only if unit extraction costs increase with the passage of time. CLASSIFICATION OF DEPOSITS AND CONSUMING AREAS

Coal deposits are distinguished on the bases of location and physical characteristics. Two deposits may be considered physically identical if their extraction functions are identical, and may be treated as a single deposit for purposes of analysis if they are identically located. Since coal is a bulky commodity, and transport costs are a large proportion of total costs, two deposits which have different locations must be treated separately even though they may be physically identical. Likewise, deposits which are identically located must be treated separately if their extraction functions differ. The first step in the classification of deposits is made on the basis of location. The United States is divided into n geographically contiguous districts to allow an unambiguous definition of interdistrict transportation. Deposits within districts are distinguished on the basis of physical characteristics. There may be any number of deposits within a district, but to ease the burden of presentation, let us assume that there are two deposits 2

An outstanding example is Harold Hotelling, "The Economics of Exhaustible Resources," Journal of Political Economy, XXXIX (April 1931), pp. 137-175. 14

A Short-Run Model for the Coal Industry within each district. This is consistent with the distinction between underground and surface deposits that is used for the empirical applications of the model, though it easily can be extended to include a larger number. The consumption of coal is distinguished solely on the basis of location. Coal is demanded and consumed in each of the n districts. A homogeneous output is secured from all of the deposits which extract coal, and the consumption requirements of a district can be met by deliveries from any of the deposits. There are 2n2 possible deliveries of coal from the 2n deposits to the n consumption locations. The number of possible deliveries increases very rapidly as the number of districts is increased. Though conceptually n can be any finite integer, practical considerations limit its size for any application of the model. Special methods are available to facilitate the solution of the model, and the effective limitation on the size of n is imposed by data availability rather than computational feasibility. Exports and imports are easily introduced into the model even though its geographical classification is limited to the United States. Exports are additions to the demands of the districts from which they leave the United States; imports are regarded as deliveries from fictitious outside districts with "extraction" costs equal to their total costs at ports of entry and with transport costs computed from the districts in which the ports of entry are located. DEMAND REQUIREMENTS

A fixed quantity of coal is demanded in each district. The spatially distributed demands are a part of the data, and are determined outside of the model. Price elasticities of demand are zero; the demands may be regarded as sums of derived, or technologically determined, quantities required by the economy to meet prescribed production and export levels. An alternative interpretation, which is used for the applications of the model, regards the demands as historically determined quantities. In any case, the consumers of coal will select those deliveries which can be purchased at the lowest prices. Beyond this, consumers are completely indifferent as to the source of the coal which satisfies their demands.

15

The Efficiency of the Coal Industry The demands must be met exactly. Therefore the demand requirements of the model are given by a system of n equations : 2

2 4 = 4 0=1,2,...,») (i) 1 i=l where dj is the demand for coal in the yth district, and x-j is the quantity of coal which is extracted from the /rth deposit in the ¿th district and delivered to the 7th district. Individual delivery levels may be either positive or zero. The demands must be met from current output since the model is static and does not allow the accumulation or depletion of inventories. Total current output therefore equals total demand. CAPACITY RESTRICTIONS

Each deposit is subject to a capacity restriction which states the maximum amount of coal which can be secured from that deposit during the short-run period. The actual output of a deposit may be at any level from zero to its capacity. Capacities are described by nonpositive and deliveries by nonnegative numbers in accordance with linear programming convention. The capacity restrictions of the model are given by a system of 2n weak inequalities : - 2 4 ^ k i (A = l, 2; i'=l, 2 n) (2) j=1 where M* is the nonpositive capacity of the M i deposit in the z'th n

district. 3

The sum of the deliveries from a deposit ]> xij j=1 equals its total current output. EXTRACTION COSTS

As each producible commodity has a production function which relates its output to the inputs used in its production, each coal 3

If capacities were regarded as nonnegative numbers, the inequality signs would be reversed, and the capacity restrictions would be in a more easily recognized form :

2 A é $ 1-1 which states that the output of a deposit cannot exceed its capacity. This form is used at a later stage when the methods for the computation of the minimum cost solutions are presented.

16

A Short-Run Model for the Coal Industry deposit has an extraction function which relates its output of coal to the inputs used in its exploitation. Since nature provides coal deposits under a diversity of conditions, there is a separate extraction function for each deposit. The physical characteristics of the various deposits are indirectly described by their extraction functions. Extraction functions, as well as production functions, can be expressed in very general terms, but a numerical analysis requires that their shapes be specified. Variable inputs are assumed to be applied in fixed proportions in the exploitation of a deposit during a short-run period. These proportions are not necessarily the same for different deposits, and may vary for a given deposit from one short-run period to the next. The short-run extraction function for a deposit can be described by the cost of the inputs necessary for the extraction of a unit of coal since inputs are applied in fixed proportions and input prices remain constant during a short-run period. The total variable extraction cost for a deposit is a linear and homogeneous function of the output of that deposit: =

(* = 1,2; »=1,2,...,»)

(3)

3=1

where G* is the total variable extraction cost for the hth deposit in the z'th district, and af is the constant unit cost for that deposit. Costs which are invariant with respect to the output of a deposit —the so-called fixed costs—are not of interest at this stage since they only add a constant term to the total cost of meeting the demands, and have no influence upon the minimum cost solutions. Unit extraction costs are separated into two components: those unit costs which are uniform for every deposit (/S), and those which differ for at least two deposits (e1-): = £+

(h= 1,2; i=\,2,...,n)

(4)

The contribution of the components to total cost depends upon the level of total output, but is invariant with respect to its distribution among the various deposits. Total output equals total demand, and is constant for any specified set of demands. Therefore, the /? components have no influence upon the minimum 17

The Efficiency of the Coal Industry cost once the demands are specified, and may be omitted for the determination of minimum cost solutions.4 Normal royalties, as contrasted with the differential royalties or economic rents determined in the dual (price) program, are treated as uniform payments for each unit of coal extracted. Even those deposits which earn a zero differential royalty receive the normal royalty.5 Normal profits are also regarded as uniform unit costs. The exclusion of normal royalties, normal profits, and other costs which may be contained in ¡3 is not necessary, but is convenient since these costs then need not be estimated for an application of the model. The only extraction costs which need be considered for the determination of the minimum cost solutions are those given by the e'1 terms. The total included extraction costs for a deposit are given by a linear and homogeneous function: =

(A= 1, 2; i = 1,2,...,n).

(5)

3= 1

Henceforth, "extraction costs" will mean only those costs included in (5) unless some other meaning is indicated. The appropriateness of the particular shapes of the extraction and cost functions which have been assumed is an empirical question which cannot be answered on the basis of theoretical analysis. If functions of these shapes provide reasonably accurate approximations of actual situations, they may be used; if not, 4

Substituting (4) into (3) and summing for all deposits: 2

n

2

n

n

2 2G? = 2 2 0 8 + 4 ) 2 4 . By a rearrangement of terms, total extraction cost can be expressed as the sum of its two components:

2 2 g? = px+i

h-l¡-1

2 *?2 4

h=l i-1

l-l

where AT=2»-i 2?=i 2?=i xti ' s t o t al output. The uniform cost, fiX, is invariant with respect to variations of the outputs of the various deposits. 6 Similar distinctions have been made by Alfred Marshall, Principles of Economics (8th ed.; New York, 1920), Book V, chap, x, pp. 4 3 8 ^ 3 9 ; and John E. Orchard, "The Rent of Mineral Lands," Quarterly Journal of Economics, XXXVI (February 1922), pp. 290-318.

18

A Short-Run Model for the Coal Industry their use is inadmissible. The appropriateness of these functions is considered at a later stage. 6 TRANSPORT COSTS

T h e unit cost of a delivery is the sum of its unit extraction and unit transport costs. There is a separate unit cost for each of the 2n 2 possible deliveries : 4 = 4 +%

(h = \ , 2 ; i , j = \ , 2 , . . . , n )

(6)

where c[| is the unit cost of a delivery to the 7th district of coal which was extracted from the M i deposit in the ¿th district, and $ is the unit transport cost of this delivery. Unit transport costs are determined outside of the model, and are invariant with respect to the size and distribution of the deliveries which are made. Transport costs are not calculated for point-to-point shipments. Coal extraction and consumption are widely distributed, and some degree of aggregation is necessary for any practical application of the model. T h e nature of the necessary aggregation is indicated in the following diagram for a simple system with three districts, each of which contains one deposit with two extraction locations (E,) and two consumption locations (C,). T h e possible deliveries from District 1 to District 2 are connected by straight

Ez

2

6

3

See below, pp. 55-60. 19

The Efficiency of the Coal Industry lines with arrows indicating the direction of movement. The single delivery x12 is actually a complex of four point-to-point deliveries. The costs of these individual deliveries are averaged to obtain a transport cost for the complex delivery. The assumption that the unit transport cost of a complex delivery is constant implies that the proportions of its components do not change. Since extraction and consumption cannot be assigned to points, the transport costs of intradistrict deliveries are not zero. The individual components of the intradistrict delivery are also outlined in the diagram. If coal is consumed at the mines, two additional consumption locations with zero transport costs must be averaged together with the other four deliveries to obtain the unit transport cost for this composite intradistrict delivery. Deliveries from the deposits in District i to the consumption locations in District j clearly are not the reverse of deliveries from the deposits in District j to the consumption locations in District i, and there is no reason to expect to equal The accompanying diagram easily could be extended to demonstrate that the transport costs for coal extracted from the underground and surface mines within a district will differ if the mines are differentially located or differentially weighted. THE ALLOCATION PROBLEM

Any set of nonnegative delivery levels which satisfies the demand requirements, (1) above, and the capacity restrictions, (2) above, is a solution for the model. If total demand exceeds total capacity, the demand requirements are not feasible, and the model has no solution. Otherwise, the model has many solutions. If one solution is discovered, others can be derived from it by various rearrangements of the delivery levels. An optimum solution for a linear programming problem is one which minimizes (or maximizes) a linear function of the variables. An optimum solution for the present model is one which minimizes the total costs of extraction and transportation. The linear function to be minimized is the sum of the delivery levels multiplied by their respective unit costs: » = i I t 44h=l i=l j=l 20

(?)

A Short-Run Model for the Coal Industry The complete programming problem is given by the demand requirements (1), the capacity restrictions (2), the condition that total costs (7) are to be minimized, and the condition that all delivery levels must be nonnegative. Minimum cost and efficiency are synonymous within the context of the present model. A minimum cost solution is efficient in the sense that any other solution would have a greater cost, and would therefore result in a misallocation of resources. It is of interest to note that monopoly behavior and efficiency are consistent within the framework of the present model. A monopolist would have no control over total output, and would merely minimize his costs of meeting the prescribed demands. Prices however would be completely indeterminate under monopolistic conditions; the demands have zero price elasticities, and profit maximization would dictate infinite prices. The possibility of monopoly is of little interest for the coal industry which is composed of a large number of relatively small sellers. It will be demonstrated with the aid of the dual program that an efficient solution is also consistent with perfect competition, and that prices are determinate under a regime of perfect competition. THE PROGRAMMING FORMAT

The relations of the delivery program of the model are now expressed in terms of vectors and matrices, and the program is placed within the standard linear programming format. Let us first write out the demand requirements and capacity restrictions given in (1) and (2) above: 1^

^ ^nl X

•+XÌ

\n + " " + X~„ = dn > *ì

- tfjj -x„ -xr

-X7 a ** (8a)

21

The Efficiency of the Coal Industry The first n relations are the demand equations, and the following 2n are the capacity restrictions. The positive value of a delivery level is contained in the equation for the demand which it satisfies, and its negative value is contained in the inequality for the capacity which it absorbs. Each of the 2n 2 delivery levels is contained in only two relations—each column above contains only two entries. A delivery can be described by the product of its unit vector and its level : 0" " 0 • 1

4

4 = -1

- 4

0_

o _

The unit vector for a delivery describes .that delivery at the unit level. It contains 3n elements corresponding to the 3n relations of the system. The element in the row corresponding to the demand which it meets is +1, the element in the row corresponding to the capacity which it absorbs is —1, and the remaining (3M —2) elements are zeros. The elements of the unit vectors are in the same order as the equations listed above. The unit vector for contains a + 1 in its jth row, and a — 1 in its [n + (h— l)« + i]th row. If a vector is multiplied by a scalar (number), in this case the delivery level 4 , all of its elements are multiplied by that scalar. The entire system of relations can be written in terms of vector products : -

r 0 -l

-

0" 1 0

-

V

-

0 0

0-

-d{

1 0

dn k\ A2 X

«Î1+-+ 0 0

-1 0 0_

0 -1 0_ 22

nn

0 0

> =

k k

}

-k.

A Short-Run Model for the Coal Industry A single weak inequality sign is used for all of the relations. The demands are strict equalities, but also satisfy this weaker condition. The unit vectors can be combined to form a "technological" matrix, and the programming problem written in terms of this matrix and column vectors of the delivery levels, demands and capacities: 1

0

1

0"

A

0 -1

1 0

0 0

1 0

d„ k\

0 0

-1 0

0 -1

0 0

0

0

0

-1

>

K ki ki

The technological matrix contains 3n rows and In 2 columns, the column vector of delivery levels contains 2n2 components, and the column vector of demands and capacities contains 3M components. The demand relation for the first district is obtained by multiplying the elements of the first row of the technological matrix by the corresponding elements in the vector of delivery levels, the capacity relation for the first deposit in the first district is obtained by multiplying the elements in the (n+ l)th row of the technological matrix by the delivery levels, and so on. 7 The relations of (8a) can be written in a more compact notation: AX ^ B

(8b)

where A is the technological matrix, X is the column vector of delivery levels, and B is the column vector of demands and capacities. This equation system is quite general. Solutions which contain one or more negative delivery levels 7 A 2« 2 -component column vector is the same as a (2n2 x 1) matrix with 2n 2 rows and one column. In the present case a (3m x 2n 2 ) matrix is multiplied on the right by a (2n 2 x 1) matrix. The product is a (3m x 1) matrix, or a 3m-component column vector. The rules for matrix multiplication can be found in a number of textbooks. See, for example, A. C. Aitken, Determinants and Matrices (7th ed.; Edinburgh, 1951); or Sam Perlis, Theory of Matrices (Cambridge, Mass., 1952).

23

The Efficiency of the Coal Industry are ruled out by restricting the delivery levels to nonnegative values: ( A = l , 2 ; f,>=l,2,...,n)

(9a)

A negative delivery level would suggest that coal was being taken from a consumption location and being replaced in one of the deposits. This, of course, is meaningless in the present context. The relations of (9a) can also be written in a matrix notation: X ^ 0,

(9b)

which states that every element of X must inequality. Here 0 is a 2n 2 -component column entirely of zeros. Finally, the total cost function is written as 2« 2 -component row vector of the unit costs vector of delivery levels:

satisfy the weak vector composed the product of a and the column

x

\i

z — (C n ,...,

Cfj,..., C%n)

l

u

(10a)

where the components of the row vector of unit costs are listed in the same order as those of the column vector of delivery levels.8 This function can also be written in a matrix notation: * = C'X.

(10b)

The prime superscript is used to denote a row vector. In the compact matrix notation, the programming problem is to find a 2n 2 -component column vector X satisfying (8b): AX > B, such that (9b): X ^ 0, 8 A 2n 2 -component row vector is the same as a ( 1 x 2 n 2 ) matrix with one row and 2n 2 columns. Here a (1 x 2n 2 ) matrix is multiplied on the right by a (2M2 X 1) matrix, and the product is a (1 x 1) matrix, or a scalar (number).

24

A Short-Run Model for the Coal Industry and (10b): z = C'X = minimum. The derivation of the dual problem is facilitated by placing the original problem in this format. THE DUAL PROBLEM

A minimum cost solution is consistent with the behavior of a monopolistic firm which desires to minimize its costs of meeting the prescribed set of demands, but we are unable to determine a set of prices which is consistent with monopolistic behavior. Now that the model has been placed in the programming format, we are in a position to ask if the efficient (minimum cost) solution would be realized if allocation were guided by competitively determined prices. The duality theorems of linear programming provide an affirmative answer. Though the delivery program does not explicitly account for prices, another system which does, technically known as its dual, can be derived from it. A unique set of delivered prices and differential unit royalties are determined in the dual system, and will be demonstrated to be consistent with both the efficient solution and perfectly competitive pricing. Hence, the efficient solution is demonstrated to be consistent with conditions of perfect competition as well as with those of monopoly. The dual programming problem for the system given by (8b), (9b), and (10b) is to find the 3«-component row vector P satisfying P'A ^ C

(11)

P i 0

(12)

y = P'B = maximum

(13)

such that and where y is a scalar, and A, B, and C have the same interpretation as in the delivery program. The interpretation of the dual system of course varies from one programming problem to another. These two systems are clearly interrelated. The original system contains 3n relations and 2m2 variables; its dual contains 2n 2 relations and 3n variables. The relations of one system are 25

The Efficiency of the Coal

Industry

associated with the variables of the other. Let the variable of the dual system which is associated with the /th demand relation be interpreted as the delivered price of coal in the jth. district, and be written as pj. Let the variable associated with the capacity relation for the hth deposit in the /th district be interpreted as the differential unit royalty earned by that deposit, and be written as r f . Now P=(p1,..., pn, r\,..., r\, rf,..., r% and the 2« 2 relations given by (11) can be written in a more specific form: Pj-rt^cti ( ¿ = 1 , 2 ; / , ; = 1,2,...,«). (14) These relations state that delivered prices net of differential unit royalties cannot exceed unit costs for any of the possible deliveries. The basic duality theorem 9 states that if a finite solution exists for either system, a finite solution exists for the other, and minimum z = maximum y.

(15)

The minimum total cost equals the maximum value of the linear function of the dual system. The equilibrium conditions of perfect competition are satisfied by the dual system. There is a single delivered price in each district, and a uniform unit royalty is earned by every unit of coal which is extracted from a particular deposit. Deliveries which are made earn a zero unit profit, and deliveries which are not made would earn a negative profit. Deposits which are fully utilized generally earn positive unit royalties, and those which are not earn zero unit royalties. The competitive results are derived from special properties of dual systems: (1) if the inequality holds for the ¿th relation in the optimum solution for one system, the /th variable of the other equals zero, and (2) if the /th variable of one system is positive, the /th relation of the other is an equality. 10 9 This duality theorem has many equivalent forms. Proofs are given by D. Gale, H. W. Kuhn, and A. W. Tucker, "Linear Programming and the Theory of Games," Activity Analysis, pp. 317-329; Charnes, Cooper, and Henderson, Linear Programming, pp. 72-75; and G. Dantzig and A. Orden, Notes on Linear Programming: Part II, "Duality Theorems" (Santa Monica; Rand Corporation, April 10, 1953). 10 Dantzig and Orden, "Duality Theorems," p. 3.

26

A Short-Run Model for the Coal Industry Applying this property to the variables of the delivery system and the relations of its dual, Pj-ri = 4

if

xl > 0

(16)

Pi-r*f > 0

if

r'l = 0

if

and

- f 4 = k) j=i

(18)

- f 4 > kf. (19) y=i Those capacities which are fully utilized earn a positive unit royalty (18), and those which are not earn a zero unit royalty (19). 12 The maximized function of the dual system, given in the general matrix notation in (13), is the sum of the demands multiplied by the delivered prices and the capacities multiplied by the unit royalties: 11 Relations (16) and (17) are exact if the optimum solution of the delivery system is not degenerate (see p. 31), and the included delivery with the lowest cost from each deposit is unique. The dual equality will hold for one or more * ( i = 0 if degeneracy occurs. A similar qualification is applicable for (18) and (19). 12 If total demand equals total capacity, all deposits are fully utilized and all royalties positive. The differentials between the unit royalties of the various deposits would be determinate, but their absolute levels, and hence prices, would not. Unit royalties and delivered prices are uniquely determinate if (19) holds for at least one of the deposits, i.e., if total capacity exceeds total demand.

27

The Efficiency of the Coal Industry

y=

I 1M + y=

I r=i A=I

(20)

and can now be interpreted as total revenue net of royalty payments. Total royalty payments are the sum of the unit royalties multiplied by the outputs of the various deposits. Equation (20) expresses royalty payments as the sum of the unit royalties multiplied by the capacities, but this is equivalent to multiplying the unit royalties by the outputs since nonzero royalties are only earned by those deposits which are fully utilized (for which output equals the positive value of capacity). The royalty term of (20) is negative since all of the capacities are negative. In perfect competition, the solution obtained through the maximization of net revenue is the same as the solution obtained through the minimization of total cost. The optimum solutions of the original delivery system and its dual price system together provide a complete description of a perfectly competitive equilibrium. A METHOD OF SOLUTION

A number of different methods can be used for the solution of linear programming problems. Most of the available methods are iterative. An initial solution which satisfies the requirements and restrictions of the problem is selected, and then is examined to determine if the corresponding value of the linear function can be reduced 13 by altering the values of some of the variables. If it can, the necessary alterations are made, a new solution is formed and the procedure is repeated until no further reduction is possible. The solution reached at this point is an optimum solution, and the problem is solved. The best solution method for any particular problem is the one which is likely to involve the least amount of computational work to reach an optimum solution. The technological matrix for the present model is a very special type of matrix, each column of which contains only two nonzero entries, a + 1 and a — 1. It is identical with the technological matrix for the special 13

If the problem requires the maximization of the linear function, the initial solution would be examined to determine if the value of the linear function could be increased.

28

A Short-Run Model for the Coal Industry transportation problem, and the simplified computational methods developed for this problem 14 can be applied to the present model with slight modification though there are significant conceptual differences between this and the present problem. The solution method for the present problem is illustrated for a hypothetical example, but the method is quite general, and can be used for problems which contain many more relations and variables than are contained in the example. The demands and capacities for the example are listed in Table 1. There are five districts all of which demand and three of which can extract coal. The capacity relations for the deposits with zero capacities can be omitted from the problem. For convenience the remaining six capacities are renumbered with subscripts from one through six. Capacities now are regarded as positive numbers in order to simplify the computations. Table 1.

Demands and capacities for the hypothetical system Demands

Capacities

¿! = 200

k\ = ¿, = 125

¿ 2 = 100

= Ä2 = 200

*1 = *5= 20

¿3= 50

kl =fe3= 500

¿§=Äe=ioo

¿ 4 = 200

k\

0

k\=

0

¿ 5 = 150

k\

0

¿1=

o

k\ = kt=

30

The computational methods for transportation problems are applicable to problems for which the capacity restrictions are equalities, and for which total capacity equals total demand. The capacity restrictions of the present model are weak inequalities, and total capacity is likely to exceed total demand. However, a simple transformation can be used to make the present model 14

George B. Dantzig, " Application of the Simplex Method to a Transportation Problem," Activity Analysis, pp. 359-373. A somewhat less mathematical explanation is given by A. Charnes and W. W. Cooper, "The Stepping Stone Method of Explaining Linear Programming Calculations in Transportation Problems," Management Science, I (October 1954), pp. 49-69.

29

The Efficiency of the Coal Industry satisfy the conditions for transportation problems. Define a demand for a fictitious sixth district which equals the excess of total capacity over total demand: = l k; - J dj != 1 7=1

(21)

6

X

= i= 2 1 i6

A "delivery" from the ¿th deposit to the sixth district is the amount of unused capacity at the ¿th deposit. This transformation adds six variables and one equation to the system, and converts the capacity restrictions into equalities: *i=t*,7 (1 = 1,2,..., 6) (22) j=i The complete system contains twelve equations and thirty-six variables. The demand and capacity equations are presented in a special tabular form in Table 2. The first six rows contain the capacity restrictions if equality signs are placed between the sixth and seventh columns, and plus signs are placed between the other columns. Likewise, the first six columns contain the demand equations if equality signs are placed between the sixth and seventh rows, and plus signs between the other rows. The unused capacities in the sixth column serve to equate the sums of the row and column totals. The unit costs for the thirty-six possible deliveries are listed in Table 3. The unused capacities are assumed to be costless. The total cost for any solution is obtained by multiplying its delivery levels by the appropriate unit costs and summing. Any set of nonnegative values for the delivery levels which satisfies both the demand requirements and the capacity restrictions is a feasible solution for the programming problem. A feasible solution which does not contain more than eleven deliveries with positive levels is a basic feasible solution. The equality of row and column totals signifies that one of the equations is a linear combination of the others. The system contains eleven independent equations which usually can be solved for 30

A Short-Run Model for the Coal Industry eleven variables with nonzero levels. A basic feasible solution which contains less than eleven variables with nonzero values— the problem of degeneracy—is possible, but not common, and is ignored. 15 Table 2.

Relations of the hypothetical system

Row

Column 1

1 2 3 4 5

2

6

X

«1

X22 x 32 X 42 x 52 x62

7

200

100

X 31 x

il

Table 3.

3

4

5

X 13 X 23 x 33 X

X 25 X 3S x 45 X 5S X

^26

«3

xlt x 24 x 34 x a X S4 x 64

50

200

150

375

43

•*5 3

x

7

6

125 300 500 30 20 100

65

1075

Unit costs for the hypothetical system Columns

Rows

1

2

3

4

5

6

1 2 3 4 5 6

2.00 2.60 3.20 1.30 2.00 2.80

2.50 1.90 2.60 1.90 1.20 2.10

3.30 2.40 2.40 2.60 1.70 1.80

2.20 4.30 2.70 1.50 3.60 2.20

5.50 3.00 3.60 4.80 2.30 3.00

0 0 0 0 0 0

The selection of an initial basic feasible solution is not difficult, but care should be used in its selection since the number of iterations necessary to reach the minimum cost solution can be reduced substantially if the initial solution is a close approximation ls If degeneracy is encountered, the powerful £ technique can be utilized to circumvent it. See Charnes, Cooper, and Henderson, Linear Programming, pp. 62-70.

31

The Efficiency of the Coal Industry of the minimum cost solution. A set of simple rules has been devised to allow the selection of such an initial solution. 16 Examine Table 3, and select a unit cost which is both a row minimum and a column minimum disregarding the zero costs in the sixth column. 17 There are a number of these minimum costs in the present example, and the computations may begin with any one of them. Let us start with the delivery from the fourth deposit to the first district. Its cost is a minimum for both the fourth row and the first column. Now select the largest possible level for this delivery. It can be increased until either the demand of the first district is fully met, or the capacity of the fourth deposit is fully utilized. The level of x41 is selected to equal the smaller of these values: x41 = min (dlt k4) = min (200, 30) = 30. The level of x 4 1 is set equal to thirty units, and the demand of the first district and the fourth deposit are reduced by the same amount. The fourth row of Table 3 can be omitted from further consideration since the capacity of the fourth deposit now equals zero. Another minimum value is selected from Table 3 after its fourth row has been deleted, and the procedure is repeated. Let the delivery from the fifth deposit to the second district be the second to be introduced. Set x 5 2 equal to twenty units: x 5 2 = min (100, 20) = 20, 16 The northeast-corner method for the selection of an initial solution which is used by Dantzig, "Application of the Simplex Method," and Charnes and Cooper, "The Stepping Stone Method," gives no consideration to unit costs, and is usually inefficient because of the large number of iterations necessary to reach an optimum solution. The author solved his programming problems using rules quite similar to those described in the text. This method has been described formally by H. S. Houthakker, On the Numerical Solution of the Transportation Problem (Stanford University, Department of Economics, Technical Report No. 15, December 23, 1954). 17 Houthakker gives the corresponding delivery the descriptive title of "mutually preferred flow," p. 2.

32

A Short-Run Model for the Coal Industry reduce the demand of the second deposit by twenty units, and strike the fifth row of Table 3. The unit cost of # 6 3 is also a minimum value. Set x 6 3 equal to fifty units: x6Z = min (50, 100) = 50, reduce the capacity of the sixth deposit by fifty units, and strike the third column of Table 3. The relevant information for the selection of the initial solution after these three deliveries have been introduced is summarized in Table 4 which contains the unit costs for the remaining rows and columns together with the reduced capacities and demands in the seventh column and row. Two rows and one column have been eliminated in the three steps thus far performed. Table 4. Relevant unit costs and reduced capacities and demands after selection of three deliveries for the initial solution Columns Rows

1

2

4

5

6

7

1 2 3 6

2.00 2.60 3.20 1.30

2.50 1.90 2.60 1.90

2.20 4.30 2.70 1.50

5.50 3.00 3.60 4.80

0 0 0 0

125 300 500 50

7

170

80

200

150

375

975

— 30,

X52 — 20,

x^g — 50

Following the same procedure, the remaining steps are as follows: let x 6 1 = 50 and delete the sixth row * n = 120 and delete the first column X22 = 80 and delete the second column = 5 and delete the first row #34 = 195 and delete the fourth column x 2 5 = 150 and delete the fifth column x 2 6 = 70 and delete the second row x 36 = 305 and delete the third row. Eleven steps are necessary to select the initial solution. One equation is satisfied by each step; the initial solution is a basic 33

The Efficiency of the Coal Industry feasible solution. The sixth column is automatically satisfied since all of the rows are satisfied. In this particular example there is at least one minimum value at each step. If a situation should arise in which one does not exist, any of the deliveries contained in the rows and columns which have not been deleted could be introduced, but in order to simplify subsequent computations, an attempt should be made to select a delivery with a unit cost which is close to a minimum value. The initial basic solution is presented in Table 5. Eleven deliveries have positive, and the remaining twenty-five have zero values. The reader can easily verify this solution by checking the row and column totals. The total cost for this solution is 1,672.50 dollars. The next step is to determine if total cost can be reduced through the introduction of any of the deliveries excluded from the initial solution. The introduction of an excluded delivery would require changes in the levels of three or more of the included deliveries in order to continue to satisfy the demand and capacity equations. For example, if x6l (see Table 5) were introduced, and xXi would be reduced and increased by equal amounts to maintain row and column totals. This pattern is outlined in Table 5 by a series of plus and minus signs. Table 5.

Initial basic feasible solution

Columns Rows

1

1 2 3 4 5 6

120«

7

200

2

3

4

5

6

150

70 305

150

375

5H

80

195 30 20 50H 100

50

*64

*« = =

-r! + rt,

or ^16 =

~ri

since r 2 = 0. The same result can be derived if a larger number of included deliveries is affected by the introduction of an unused capacity. The optimum solution of the dual program can be derived directly from the minimum cost solution of the delivery program. The unit royalties are contained in the sixth column of Table 10, and the delivered prices are easily computed using relation (16): Pi = p2= p3 = Pi = Ps =

c

2i + r2 = c22 + r2 = ^63 + ^6 = c3t + r9 = ^25 + ^2 =

40

2

-60 1.90 2 30 2.70 3.00.

CHAPTER III

Applications of the Model A SHORT-RUN period for the purposes of the empirical applications of the model is defined as one year. The output of the coal industry is very sensitive to cyclical movements, and is subject to large annual variations. A minimum cost solution for any particular year might be representative of the general output level for that year, but might prove to be quite different from a solution for a year with a higher or lower general output level. The model is separately applied for three postwar years which are representative of three different levels of annual output. The selected years are 1947 when a record output of 631 million tons was extracted, 1949 when 430 million tons were extracted, and 1951 when 534 million tons were extracted. The historical demands, capacities, and unit costs for these years are the data used for the empirical implementation of the model. The model is not projected to future periods, and it is not necessary to predict its data. The efficient solutions give the delivery levels which would have prevailed if total costs had been minimized in these years. OUTPUT INDEX

A homogeneous output of coal is assumed throughout the theoretical development of the model, but this assumption is not satisfied if the data and variables are expressed in tonnage units. A ton of North Dakota lignite is hardly the equivalent of a ton of West Virginia bituminous. Coals are classified by a number of characteristics other than weight. Coal analyses are made to determine average heating values, and average amounts of moisture, volatile matter, fixed carbon, ash, and sulfur. 1 A 1 See A. C. Fieldner, W. E. Rice, and H. E. Mason, Typical Analyses of Coals of the United States, Bureau of Mines, " Bulletin " 446 (Washington, D.C., 1942).

41

The Efficiency of the Coal Industry committee of the American Standards Association has grouped coals with respect to (1) size, (2) heating value, (3) amount of ash, (4) temperature at which ash softens, and (5) amount of sulfur. 2 The number of different grades of coal is very large if all five of the factors mentioned above are considered. An output index, or one dimensional measure, must be used to allow comparison of coals extracted from different deposits. Tonnage figures are generally used by the Bureau of Mines and other investigators, but this practice is subject to severe criticism. Consumers do not judge coals on the basis of weight. Heating value is the most important single characteristic of coal from the consumers' viewpoint, and therefore is used as an output index for the applications of the model. Most demands for coal may be expressed in terms of British thermal units (B.t.u.). 3 The demands, capacities, and unit costs were estimated in thousands of tons and dollars per thousand tons, and then converted to heating value units and dollars per heating value unit before the model was applied. A heating value unit is defined as 10 10 (ten billion) B.t.u. Once this transformation is made, a thousand tons of West Virginia bituminous which has an average of 2.790 heating value units is the equivalent of approximately two thousand tons of North Dakota lignite which has an average of 1.398 heating value units per thousand tons. Coking coals must possess a combination of properties, and cannot be described in terms of heating value alone. The present applications of the model therefore exclude coking coal. The capacities and demands are net of coking coal output and consumption. Other special purpose coals have not been omitted, but are of less importance. The major consumers of coal, particularly electric utilities; distinguish between coals mainly on the basis of heating value content. Pennsylvania anthracite is also excluded from the present analysis. Hard and soft coal are separate commodities with different properties and uses. An insignificant amount of anthracite is extracted outside of Pennsylvania, and its inclusion 2 Cited by Waldo E. Fisher and Charles M. James, Minimum Price Fixing in the Bituminous Coal Industry (Princeton, 1955), p. 119. 3 A British thermal unit is the quantity of heat required to raise the temperature of one pound of water one degree Fahrenheit.

42

Applications of the Model can do little harm. Throughout the remainder of this study, unless otherwise indicated, the term coal is used to denote only those coals which are included for the applications of the model. CLASSIFICATION OF DEPOSITS

Deposits are classified by both location and method of mining in order to group together mines with similar extraction and transport costs. The continental United States is divided into fourteen geographically contiguous districts (see Table 11). Each district is composed of one or more integral states; available empirical information does not allow a finer breakdown. Coal is consumed in all fourteen districts, and extracted in the first eleven. Underground and surface deposits are separated in each of the eleven districts in which extraction takes place. These two methods of mining are quite distinct from both technical and cost viewpoints.4 The model thus contains twenty-two deposits and fourteen consumption locations. A separate unit extraction cost was computed for each deposit. The effectiveness of the classification in grouping together individual mines with similar extraction costs is considered at a later stage.5 The various deposits account for widely different proportions of total coal output (see Table 11). Approximately three quarters of the 1951 output was extracted from the underground deposits; more than one half was extracted from the underground deposits in the first three districts. Three districts (8, 10, and 11) each accounted for less than 1 percent of the total output. THE DATA

The data of the model are the historical demands, capacities, and unit costs. These were estimated separately for 1947, 1949, and 1951, actual figures being used for each year. The procedures described below have been used for each of the three years unless otherwise indicated.6 4 Surface mining is conducted in the open. The overburden of earth is removed with large power shovels and draglines, and the coal is then dug from an open pit. Surface mining generally requires considerably less labor and more capital equipment than underground mining. 6 See below, pp. 54-55. 6 A detailed description of the sources and rectification methods used to obtain the basic data is given in Appendix A.

43

The Efficiency of the Coal Table 11.

Industry

Coal districts and relative coal outputs in 1951 Percentage of 1951 Output

District 1 2 3 4

5 6 7 8 9 10 11 12

13 14

Underground

Surface

Total

10.00 28.80

5.77 3.30

15.77 32.10

17.25

3.02

20.27

1.78 3.75

.57 4.96

2.35 8.71

9.00

6.04

15.04

.32

1.55

1.87

.03

.38

.41

1.92

.65

2.57

.66 .16

.08 .01

.74 .17

73.67

26.33

100.00

Pennsylvania* and Maryland West Virginia* Virginia,* Kentucky,* and District of Columbia Alabama,* Tennessee,* Georgia,* North Carolina,* South Carolina, Florida, Mississippi, and Louisiana Ohio* Illinois,* Indiana,* and Michigan* Iowa,* Missouri,* Kansas,* Arkansas,* Oklahoma,* and Texas North Dakota,* South Dakota,* and Nebraska Montana,* Wyoming,* Utah,* and Idaho Colorado,* New Mexico,* Arizona,* California,* and Nevada Washington* and Oregon* Maine, Vermont, New Hampshire, Rhode Island, Connecticut, and Massachusetts New York, New Jersey, and Delaware Minnesota and Wisconsin

* Denotes states in which coal is extracted. Source: Output percentages were computed from information given in "Coal—Bituminous and Lignite," preprint from Bureau of Mines, Minerals Yearbook, 1951. Demands. T h e demands for coal in the various districts (see Table 12) are the actual consumption levels, including exports allocated to the districts from which they left the United States and excluding coal used for coke, for each of the three years.

44

Applications of the Model Five of the districts demanded more than 100,000 units in 1951, five demanded 40,000 to 100,000 units, and the remaining four demanded less than 40,000 units. Consumption is more widely distributed over space than extraction. Four districts extracted more than they consumed in 1951, and ten consumed more than they extracted. The first two districts extracted 48 percent of the total output, but accounted for only 17 percent of the total demand. Districts 12-14, on the other hand, extracted no coal, but accounted for 18 percent of total demand. Interdistrict deliveries are of considerable importance for the coal industry. Capacities. The short-run capacity for a deposit is defined as the maximum amount of coal which can be extracted from it in a given year with the labor and equipment available during that year. The 280-day rule which was suggested by the Coal Committee of the American Institute of Mining and Metallurgical Engineers has been used to estimate capacities for underground deposits.7 Table 12. District 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Totals

Demands for coal: 1947, 1949, and 1951 (10 1 0 B.t.u.) 1947

1949

1951

221,400 66,276 139,708 107,089 165,421 283,305 62,773 15,763 28,548 18,108 10,522 60,186 123,221 47,005

144,947 47,751 97,422 67,497 127,682 223,345 43,906 10,527 22,258 10,535 7,150 48,029 90,878 37,334

133,031 45,099 149,732 68,633 140,904 242,393 48,607 11,497 21,507 11,192 7,985 52,293 102,105 42,901

1,349,325

979,261

1,077,879

Sources: See Appendix A.

A 280-day capacity is computed by extrapolating average output per day, for the days the deposit was active, to 280-days. ' Bureau of Mines, Minerals Yearbook, 1935, pp. 631-632.

45

The Efficiency of the Coal Industry The computational formula is 280

k

i =

— i

a

where k{ is the positive capacity of the z'th underground deposit for a given year, qi is the actual output of the deposit for that year, and ai is the average number of days the deposit was active. Capacity is defined to equal zero if the number of days active equals zero. The average annual number of days active for all mines since 1900 has varied from a low of 142 in 1922 to a high of 278 in 1944. The former was a year of widespread strikes, and the latter a year of war production. The averages for all underground mines are 238 days for 1947, 156 for 1949, and 203 for 1951. None of the underground deposits were active for as many as 280 days during these three years, though a number of individual mines exceeded the 280-day limit. The 280-day capacities appear to provide reasonably good measures of maximum output levels. They are not so high that they can never be attained, as is witnessed by the 278 days active in 1944, but on the other hand, they are not so low that they are easily exceeded. The Bureau of Mines formerly presented capacity figures based on an assumed 308 days active which corresponds to a 6-day work week throughout the year, but it has discontinued this method, and now only computes 280-day capacities. The AIMME 280-day estimate was constructed to account for delays due to breakdowns, falls of roof, power failures, and the seasonal character of the demand for coal.8 These delays prevent a 6-day work week throughout the year in the average mine. The 280-day estimate corresponds to a 5.4-day work week. Capacities for the surface deposits were computed in a similar manner except that capacity operation was assumed to be 260 days, rather than 280. Surface mining is technically quite different from underground mining, and this lower rate of operation appears to provide a better measure for the surface deposits. Surface mining, being conducted in the open, is more sensitive to inclement weather, and the 260-day capacity accounts for this and other special factors. None of the surface deposits 8

Minerals Yearbook, 1935, p. 632.

46

Applications of the Model have been active for more than 260 days in the postwar period, though several have come close to this limit. All surface mines were active an average of 202 days in 1947, 165 in 1949, and 199 in 1951. T h e estimated capacities for the underground and surface deposits for the three years encompassed by the present study are given in Table 13. More than half of the total capacity is Table 13.

District

Capacities by districts and methods of mining: 1947, 1949, and 1951 (10 1 0 B.t.u.) 1947

1949

1951

Underground Deposits 1 2 3 4 5 6 7 8 9 10 11 Totals

225,598 381,733 270,831 49,918 64,661 165,348 12,828 690 36,690 20,424 1,388

234,940 436,395 302,864 55,505 62,998 161,249 10,500 728 36,894 18,890 2,426

175,930 426,845 305,632 41,056 55,603 174,484 7,159 419 32,138 13,564 2,263

1,230,109

1,323,389

1,235,093

Surface Deposits 1 2 3 4 5 6 7 8 9 10 11 Totals Total capacities

142,982 97,282 40,506 7,613 52,341 74,997 27,682 3,365 7,382 927 609

101,479 76,706 42,828 9,648 65,451 78,565 26,890 3,952 6,635 1,630 520

92,215 56,931 44,037 9,075 66,683 70,132 23,087 4,568 8,599 1,372 290

455,686

414,304

376,989

1,685,795

1,737,693

1,612,082

Sources : See Appendix A.

47

The Efficiency of the Coal Industry located in Districts 1 (Pennsylvania and Maryland) and 2 (West Virginia). Total capacity increased from 1,685,695 units in 1947 to 1,737,693 in 1949, and then declined to 1,612,082 units in 1951. The total capacities are considerably larger than the total demands, and the coal industry possesses a substantial amount of unused capacity. The total capacity of the coal industry is at about the same level today as in 1920. Bureau of Mines capacity figures, converted to heating value units, show a peak capacity of 2.3 million units in 1923, a decline after 1923 to a low of 1.5 million units in 1933, and a fairly steady increase after 1933 though total capacity has not returned to its 1923 peak. Total surface capacity increased rapidly both in absolute amount and in relation to underground capacity in the decade and a half before 1947, and then declined after 1947. The figures for total capacities fail to reveal the diverse movements for the capacities of individual deposits. Between 1947 and 1951 the capacities of six of the surface deposits (Districts 3, 4, 5, 8, 9, and 10) increased, and the capacities of the other five decreased. The capacities of individual underground deposits likewise experienced diverse movements. The capacities of four underground deposits (Districts 2, 3, 6, and 11) increased, and those of the other seven underground deposits decreased. The capacities form a part of the data of the model, and are not explained by it. Year-to-year capacity fluctuations may not be of particular importance for an interpretation of the results of the model, but a steady increase or decrease of a particular capacity over a number of years would require special investigation to determine the long-run factors in operation. Extraction costs. Unit extraction costs were separately estimated for each of the deposits for each year (see Table 14). These include all extraction, administrative, and selling costs except royalties, depletion, and profits. Despite recent trends toward mechanization, coal mining, particularly underground mining, requires a great deal of direct labor relative to other inputs. Approximately 65 percent of the costs for underground, and 35 percent of those for surface deposits are direct labor costs. The unit cost of extracting coal from a surface deposit is less, in every case, than the unit cost of extracting coal from an under48

Applications of the Model Table 14.

District

Unit extraction costs by districts and methods of mining: 1947, 1949, and 1951 (current dollars per 10 10 B.t.u.) 1947

1949

1951

Underground deposits 1 2 3 4 5 6 7 8 9 10 11

1,503 1,329 1,391 1,891 1,414 1,238 2,471 1,659 1,343 1,704 2,855

1,722 1,487 1,561 2,169 1,563 1,444 2,824 1,654 1,532 1,890 3,022

1,825 1,562 1,631 2,044 1,608 1,435 2,947 2,254 1,611 2,103 3,592

Surface deposits 1 2 3 4 5 6 7 8 9 10 11

984 977 706 1,511 994 1,039 1,227 1,218 734 1,400 1,657

828 893 655 1,308 897 934 1,088 1,185 761 1,220 1,484

1,063 1,096 786 1,549 1,083 1,127 1,355 1,345 849 1,384 1,880

Sources: See Appendix A. ground deposit in the same district. 9 The wide range of the unit extraction costs indicates the diversity of conditions under which coal is extracted. The range for 1947 is from a low of 655 dollars ' The differential between underground and surface costs has been noted in a number of other studies. Cf. O. E. Kiessling, F. G. Tryon, and L. Mann, The Economics of Strip Coal Mining, Bureau of Mines, "Economic Paper" No. 11 (Washington, D.C., 1931); Office of Temporary Controls, Office of Price Administration, Economic Data Analysis Branch, Survey of Commercial Bituminous Coal Mines, "OPA Economic Data Series" No. 15 (Washington, D.C., n.d.); and Herman D. Graham, The Economics of Strip Coal Mining, University of Illinois, "Bulletin" No. 66 (Urbana, 1948).

49

The Efficiency of the Coal Industry per unit for the surface deposit in District 3 to a high of 2,855 for the underground deposit in District 11. The highest unit cost is 3.8 times as large as the lowest. Little can be said on the basis of extraction costs alone since the cost of a delivery includes a transport as well as an extraction cost. High transport costs can offset the advantages derived from low extraction costs. The importance of transport costs is well illustrated by the fact that the minimum cost solution for 1947 specifies the full utilization of both the surface deposit in District 3 and the underground deposit in District 11. Relative unit extraction costs are remarkably stable for the period 1947-1951. A comparison of the ranking of the unit extraction costs for the twenty-two deposits for 1947 with those separately estimated for 1951 gives a rank correlation coefficient of 0.989. 10 Costs in current dollars increased from one year to the next as a result of increased wage and price levels. The stability of the unit costs means that approximately the same deliveries are included in the minimum cost solutions for neighboring years. Solutions might be quite unstable if the relative extraction costs did not possess this stability, and individual shortrun solutions would have little significance. Transport costs. Coal is a bulky commodity of low value in proportion to its weight, and transport costs comprise a large proportion of the total costs of meeting the demands for coal. Coal is shipped by boat, truck, and rail, but railroads are the dominant means of transportation with more than 80 percent of the total output of coal shipped from the mines by rail. The average transport costs used for the implementation of the model were obtained by first estimating average railroad transport costs from information covering actual rail shipments for each of the three years, and then adjusting these to account for other modes of transportation. Transport cost information is not available for all of the 308 possible deliveries from the 22 deposits to the 14 consumption 10 The rank correlation coefficient was calculated using the formula r* = 1—6 2f di/(n3 — n), where r* is the rank correlation coefficient, n is the number of paired observations (twenty-two in this case), and the di's (¿=1, 2, . . ., 22) are the differences in rank of like observations. See Frederick C. Mills, Statistical Methods (rev. ed.; New York, 1938), pp. 374-378.

50

Applications of the Model locations. Some of these deliveries are not made. Transport costs become prohibitive for shipments over long distances. Furthermore, coal is not delivered from high cost deposits to the districts which contain low cost deposits with sufficient capacities to supply their needs. An arbitrary unit transport cost of M is assigned to those deliveries which have not been made since World War II. M is a cost large enough to exclude these deliveries from a minimum cost solution. A single set of transport costs per ton is used for coal extracted from both types of deposits, but the transport costs per heating value unit are not the same for those districts for which the average heating values of coal extracted from underground and surface deposits are not the same. The transport costs for deliveries from District 3 to District 4 are 3,482 dollars per ton for both types of coal. Underground coal extracted in District 3 contains 2.708 and surface coal 2.536 heating value units per thousand tons. Therefore, the transport costs per heating value unit for deliveries to District 4 are 1,286 and 1,373 dollars for underground and surface coal respectively. The unit costs of the deliveries of coal from the deposits to the consumption locations are sums of unit extraction and unit transport costs. The unit costs of all deliveries, other than those assigned a transport cost of M, are presented in Table 15. The demands (Table 12), capacities (Table 13), and unit costs (Table 15) are all of the data which are necessary for the empirical implementation and solution of the model for 1947, 1949, and 1951. Table 15.* From district

1 2 3 4 5

Unit costs of extraction and transportation: 1947, 1949, and 1951 (current dollars per 10 1 0 B.t.u.) To district

1 1 1 1 1

Underground Deposits

Surface Deposits

1947

1949

1951

1947

1949

1951

2,004 2,313 2,607 3,018 2,195

2,232 2,676 3,022 3,494 2,542

2,423 2,879 3,143 3,479 2,689

1,329 1,877 1,963 2,457 1,678

1,494 2,166 2,292 2,862 1,973

1,661 2,413 2,401 3,018 2,164

* Possible deliveries which have unit costs greater than M are omitted from this listing.

5

51

The Efficiency of the Coal Industry From district

To district

Underground Deposits

Surface Deposits

1947

1949

1951

1947

1949

1951

1 2 3 5

2 2 2 2

2,282 1,456 1,930 1,988

2,586 1,757 2,321 2,540

2,956 1,815 2,440 2,646

1,607 1,020 1,235 1,471

1,848 1,247 1,531 1,971

2,194 1,349 1,650 2,121

1 2 3 4 5 6 7

3 3 3 3 3 3 3

2,546 2,293 1,895 2,620 2,252 1,959 4,390

3,023 2,646 2,047 3,126 2,552 2,651 5,094

3,172 2,811 2,249 2,905 2,659 2,481 5,309

1,871 1,857 1,197 2,051 1,735 1,650 3,044

2,285 2,136 1,233 2,487 1,983 2,240 3,559

2,410 2,345 1,446 2,430 2,134 2,169 3,850

1 2 3 4 6 7

4 4 4 4 4 4

2,952 2,556 2,357 2,402 2,621 3,618

3,426 2,920 2,712 2,563 2,914 4,182

3,635 3,077 2,917 2,480 3,195 4,359

2,277 2,120 1,695 1,829 2,308 2,257

2,688 2,410 1,956 1,912 2,502 2,622

2,873 2,611 2,159 1,995 2,879 2,847

1 2 3 4 5 6

5 5 5 5 5 5

2,170 2,189 2,266 2,747 1,889 2,232

2,544 2,534 2,622 3,186 2,091 2,781

2,687 2,676 2,771 3,166 2,166 2,858

1,495 1,753 1,597 2,180 1,372 1,922

1,806 2,024 1,858 2,548 1,522 2,369

1,925 2,210 2,003 2,647 1,591 2,544

1 2 3 4 5 6 7 8

6 6 6 6 6 6 6 6

2,712 2,575 2,551 3,120 2,317 1,804 3,874 5,792

3,138 2,960 2,912 3,604 2,726 2,076 4,484 6,512

3,263 3,114 3,061 3,591 2,819 2,116 4,870 7,416

2,037 2,139 1,903 2,561 1,800 1,497 2,518 5,318

2,400 2,450 2,173 2,974 2,157 1,668 2,932 6,077

2,501 2,648 2,313 3,133 2,294 1,805 3,386 6,508

1 2 3 4 5 6 7 8 9 10

7 7 7 7 7 7 7 7 7 7

3,529 2,887 2,945 3,563 3,359 2,309 3,062 5,299 3,802 3,696

4,242 3,477 3,259 4,082 3,849 2,702 3,445 5,933 4,152 4,202

4,503 3,485 3,401 3,967 4,036 2,740 3,577 9,380 4,353 4,560

2,854 2,451 2,327 3,013 2,842 1,999 1,690 4,825 3,527 3,215

3,504 2,967 2,550 3,462 3,280 2,290 1,865 5,497 3,696 3,728

3,741 3,019 2,676 3,517 3,511 2,426 2,020 8,471 3,910 3,857

52

Applications of the Model From district

To district

Underground Deposits

Surface Deposits

1947

1949

1951

1947

1949

1951

4,210 4,149 3,327 3,559 4,438 2,590 3,396 4,322

4,421 4,357 3,902 3,684 4,501 3,197 3,500 4,632

3,041 3,251 2,315 2,694 2,435 2,025 2,686 3,339

3,472 3,639 2,623 3,143 2,885 2,155 2,841 3,848

3,659 3,891 3,211 3,366 2,996 2,288 2,891 3,930

1 2 3 6 7

8

9 10

8 8

3,716 3,687 2,933 3,009 3,793 2,499 3,055 3,820

2 7 9 10 11

9 9 9 9 9

6,151 5,174 2,200 3,225 4,300

7,155 6,022 2,407 3,816 4,721

7,585 6,144 2,498 3,991 5,397

5,715 3,843 1,725 2,743 2,929

6,645 4,512 1,723 3,339 3,356

7,119 4,732 1,839 3,285 3,685

2 7 9 10

10 10 10 10

5,123 4,687 3,059 2,073

5,946 5,446 3,744 2,379

7,520 5,689 3,777 2,560

4,687 3,347 2,691 1,590

5,436 3,920 3,235 1,892

7,054 4,251 3,267 1,844

7 9 10 11

11 11 11 11

7,706 3,506 4,473 3,423

9,018 3,927 4,744 3,576

8,534 4,004 5,130 4,302

6,423 3,194 3,994 2,052

7,589 3,441 4,273 2,211

7,257 3,520 4,404 2,590

12 12 12 12

3,037 2,935 3,146 3,264

3,502 3,363 3,485 3,738

3,635 3,466 3,654 3,847

2,362 2,499 2,544 2,747

2,764 2,853 2,794 3,169

2,873 3,000 2,946 3,322

13 13 13 13

2,539 2,469 2,503

2,977 2,841 3,194 2,881

3,122 2,953 3,320 2,973

1,864 2,033 2,182 1,986

2,239 2,331 2,478 2,312

2,360 2,487 2,590 2,448

14 14 14 14 14 14 14 14 14

2,453 2,463 2,548 3,029 2,191 2,609 3,903 3,378 3,442

2,858 2,839 2,935 3,500 2,427 3,037 5,022 3,562 3,883

3,042 3,020 3,125 3,470 2,495 3,111 5,076 4,248 4,070

1,778 2,027 1,900 2,468 1,674 2,298 2,548 2,905 3,121

2,120 2,329 2,198 2,868 1,858 2,622 3,484 3,127 3,392

2,280 2,554 2,382 3,008 1,970 2,795 3,604 3,340 3,594

8

1 2 3 4 5 6 7 8 9

8 8 8

8

2,810

53

The Efficiency of the Coal Industry TESTS OF THE MODEL

If one fits a regression equation to a series of historical data in order to predict the value of some variable, one usually tests its appropriateness by determining how well it "predicts" the past values of the variable. A similar testing procedure cannot be utilized for the present model since its solutions do not "predict" the actual values of the delivery levels. The solutions of the model give the values of the delivery levels which would have prevailed if total costs were minimized. A comparison of the values of the variables given by the solutions with their actual values serves to indicate how closely the coal industry approximates the norm established by the model as well as the appropriateness of the model. The model is implemented with actual data, and it is possible to test several of its assumptions. The two assumptions to be tested here are (1) that the classification of deposits groups together mines with similar extraction costs, and (2) that the extraction costs for the various deposits can be described by linear and homogeneous total cost functions. If these assumptions are not disproven, we can have a higher degree of confidence in the solutions generated by the model. Each deposit which is recognized for the applications of the model is an aggregate of many individual coal mines which are assumed to have a single extraction cost. Obviously, each of the 8,700 coal mines which were in operation during 1947 cannot be treated separately, and some degree of aggregation is a necessity. Data availability limits the districts to multiples of integral states, and the necessity of defining interdistrict transportation requires that the deposits be geographically contiguous. A large number of classifications are still possible given these restraints. We can now ask how successful the classification which has been selected is in grouping together mines with similar extraction costs. The separation of underground and surface deposits is obviously an efficient grouping, and attention can be restricted to the effectiveness of the geographical classification for each method of mining considered separately. Our question concerns the variances of the costs of the individual mines from the averages for the deposits in which they are 54

Applications of the Model included, and statistical analyses of variance and F tests are utilized to obtain an answer. Output-per-man-day figures, in heating value units, were derived on a county basis from Bureau of Mines information for 1947.11 There are 308 county observations for the underground and 200 for the surface deposits. The observations were stratified by deposits. Variances between and within deposits were computed, and F values defined as the mean squares of the variances between deposits divided by the mean squares of the variances within deposits :

i

i

where q{]- is the ¿th county observation for the yth deposit, q} is the mean of the county observations for the yth deposit, q is the grand mean, itj is the number of observations for the yth deposit, and V1 and V2 are the degrees of freedom within and between deposits respectively.12 An F table reveals that the null hypotheses that the deposit means were drawn from the same populations are rejected at the one-percent confidence level if the values of F are greater than 3.95 for the underground and 3.97 for the surface deposits. The computed F values of 15.67 for underground and 4.61 for surface deposits both exceed their critical values at the one-percent confidence level. 13 These tests demonstrate that the classification distinguishes deposits with significantly different labor productivity ratios though this is by no means the only classification which would achieve this result. Sufficient data are not available to test the effectiveness of the classification for inputs other than production labor. A basic empirical assumption of the model is that the extraction costs for each of the deposits can be described by a linear and 11 " Coal-Bituminous and Lignite," Minerals Yearbook, 1948 (Washington, D.C., 1950), pp. 271-350. 12 Analyses of variance and F tests are described in many statistical textbooks. See Paul G. Hoel, Introduction to Mathematical Statistics (New York, 1947), pp. 154-161; or A. C. Rossander, Elementary Principles of Statistics (New York, 1951), pp. 514-524. 13 The F value for the surface deposits would be increased to 5.72 if one extreme observation were omitted.

55

The Efficiency of the Coal Industry homogeneous function: gi =

j

24

where in this case (h= 1, 2), (¿=1, 2,..., 11), and 0 = 1, 2 14). The unit cost parameters e] are assumed to remain unchanged for a one year period, but are allowed to change from one year to the next. The remarkable stability of relative costs, however, suggests that this assumption is unnecessarily severe, and that the unit costs may be assumed to remain constant for periods longer than one year. The linearity assumption is made for the aggregate output of a deposit, and there is only one observation for a deposit in any given year. Time series data must be used to obtain enough observations to check the linearity assumption, but the use of time series data requires that the assumption that nothing can be said about year-to-year variations of unit costs be discarded. Difficulties resulting from the use of time series data are largely resolved by the use of a relatively short series, and by the relationship between the level of average labor cost and the extent of mechanization for underground deposits. Data availability again limits analysis to direct labor costs. Deposit outputs in tons and labor inputs in man-days have been derived from the Minerals Yearbook for the seven postwar years 1946-1952.14 A linear and homogeneous function, y = bw, is fitted to the data for each deposit, where y is labor input in man-days, w is coal output in tons, and b is a unit cost coefficient. The least squares criterion was used to determine the values of b. 15 The sums of the squared deviations of the man-day inputs from their estimated values are compared with the sums of the squared deviations from their means. The variance ratios 14 These data need not be converted to heating value units since the outputs and costs of the various deposits are not added or directly compared. 15 The selected values of b are those which minimize the sums of squares of the deviations of the man-day inputs from their estimated values.

56

Applications of the Model 7

I

y ' —

UXV

'§

1 — ê r ^ — z r ^ - , where v is the mean of the man-day observations,

I(J.-J)2

were computed, and are listed in Table 16. These ratios generally are regarded as giving the proportions of the total variances which are explained by the fitted functions. The variance ratios would equal the squares of the linear correlation coefficients if the deviations were measured from unrestricted linear functions, y = a + bzv. Since y = bw is a special case of the general linear function in which a equals zero, the variance ratios cannot exceed the squares of the corresponding linear correlation coefficients. Table 16.

Variance ratios for linear and homogeneous functions

District

Underground Deposits

Surface Deposits

1 2 3 4 5 6 7 8 9 10 11

0.967 0.756 0.707 0.843 0.850 0.914 0.977 0.745 0.909 0.993 0.707

0.873 0.926 0.946 0.605 0.902 0.839 0.883 *

0.618 0.118 0.918

* Value is negative.

The sum of the squared deviations from the general linear function cannot exceed the sum of the squared deviations from the mean of the dependent variable. The square of the linear correlation coefficient therefore is always nonnegative, and the linear correlation coefficient is always a real number. This restriction does not hold for the homogeneous functions, and their variance ratios can be negative. The smallest variance ratio for the underground deposits is 0.707. Its square root of 0.84 can be interpreted as giving a measure similar to that given by a linear correlation coefficient. The homogeneous function provides a good fit for all of the underground deposits. The deviations from the functions for the underground deposits are correlated with time. Man-days 57

The Efficiency of the Coal Industry are slightly underestimated in the early years, and slightly overestimated in the later years. The signs of the deviations are given in Table 17. A plus Table 17.

Algebraic signs of the deviations for underground mining: 1946-1952

District

1946

1947

1948

1949

1 2 3 4 5 6 7 8 9 10 11

+ + + + + + + + + + +

+ + + + + + + + + + +

+ + + + + + +

+ + + + + + +

—

1950

1951

1952

_

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

+

+

+ +

+

—

—

—

—

—

—

—

—

—

—

—

sign ( + ) indicates an underestimate, and a minus sign ( —) an overestimate. A series of underestimates is followed by a series of overestimates for ten of the eleven deposits. Man-days are underestimated for 1946-1949 and overestimated for 1950-1952 for eight of the deposits. The deviations are correlated with neither outputs nor man-days. The most likely explanation of this residual pattern lies in the increasing mechanization of underground mining, and the resulting decline in labor input per unit output. Capital is gradually being substituted for labor in the underground mines. Large changes do not take place during any single year, but a substantial amount of substitution occurs over a period of years. The percentage of underground mined coal which is mechanically loaded (see Table 18) serves as a good measure of the extent of mechanization over time. Table 18 shows a steady increase of mechanical loading for the period 1946-1952. The underground deposit in District 8 provides an exception to the general pattern. The deviations for this deposit are positive for 1946-1947, negative for 1948-1950, and positive again for 1951-1952. It followed the general pattern for 1946-1950, and then, in 1951, output, percentage of output mechanically loaded, 58

Applications of the Model and labor productivity all declined. The capacity of this deposit is only 0.5 percent of total underground capacity, and is declining. Table 18.

Percentages of underground mined coal mechanically loaded, by districts: 1946-1952

District

1946

1947

1948

1949

1950

1951

1952

1 2 3 4 5 6 7 8 9 10 11

45.3 60.2 40.9 53.0 69.5 88.1 40.5 83.0 95.1 59.6 52.9

50.1 64.7 40.6 58.8 71.7 89.0 48.3 89.6 96.2 62.7 56.8

53.8 69.2 45.3 61.9 72.7 89.9 43.9 95.0 96.5 68.0 66.9

59.1 72.5 45.2 62.5 77.6 91.8 48.4 93.9 97.2 69.7 68.0

63.9 74.2 48.3 66.7 79.4 93.1 51.8 93.6 97.8 68.2 68.4

68.5 77.0 54.5 72.0 85.2 94.9 58.3 89.5 97.9 70.0 74.6

72.0 80.3 57.8 70.0 85.5 95.3 60.8 91.8 99.0 72.6 83.5

Source: Minerals

Yearbook.

The results are good, but not quite as favorable, for the surface deposits. Seven of the eleven surface deposits have variance ratios greater than 0.83, two have variance ratios between 0.83 and 0.60, and the homogeneous function provides a poor fit for the other two. The seven deposits with variance ratios greater than 0.83, for which the function can be said to provide a good fit, accounted for 94 percent of total surface output in 1947. The residuals for the surface deposits do not follow a definite pattern, and appear to be correlated with neither time nor output. The variance ratio for the surface deposit in District 8 is negative. Labor productivity and output both increased rapidly during the period 1946-1952, and as a result the man-days observations show little variation. A somewhat similar situation exists for the surface deposit in District 10. The capacities of these two deposits represent a small proportion, about 1.5 percent, of total surface capacity. The appropriateness of the linear and homogeneous cost functions has been investigated for only the labor component of total cost. A lack of data prevents the consideration of nonlabor costs. Labor cost is a larger proportion of total cost for the 59

The Efficiency of the Coal Industry underground than for the surface deposits, and correspondingly, the linear and homogeneous labor cost function provides a better fit for the underground deposits. The assumed function is not proved correct by these computations, but it does appear to provide a good approximation for labor costs. The linear and homogeneous functions certainly are not inconsistent with available cost information.

60

CHAPTER IV

The Optimum Solutions A SEPARATE linear programming problem was formulated and solved for each of the three years, 1947, 1949, and 1951, using the data presented in the last chapter. Each problem contains fourteen demand, one unused capacity, and twenty-two capacity equations, or a total of thirty-seven equations, thirty-six of which are independent. The number of variables is much larger. The complete system contains 154 possible deliveries from the underground deposits, 154 from the surface deposits, and 22 unused capacities, or a total of 330 variables. The 140 possible deliveries which were assigned a transport cost of M are effectively excluded from the minimum cost solutions, and only the remaining 190 variables entered the computations. A programming problem which contains 36 (independent) equations and 190 variables appears quite formidable if only the numbers of equations and variables are considered. However, the solution of the present model is relatively simple if the special computational methods described in Chapter I I are utilized. 1 Our method for the selection of an initial solution which is close to the optimum proved very effective. The minimum cost solution for 1947 was reached with only one iteration, and those for 1949 and 1951 with only two. Degeneracy was not encountered at any stage of the computations; each of the minimum cost solutions (see Tables 19, 20, and 21) contain exactly thirty-six deliveries and unused capacities with positive levels. The solutions are not unique. Five deliveries which are excluded from the solution for 1947, and four excluded from those for 1949 and 1951 have opportunity costs which equal their unit costs. Any of these excluded deliveries could be introduced 1

See above, pp. 28-40.

61

The Efficiency of the Coal Industry

Table 19.

Minimum cost solution for 1947 (1010 To

1 78,418

*

48,419 99,202

99,476

42,960 64,661

142,982

221,400

66,276

66,276

40,506

139,708

7,613

107,089

52,341

165,421

35,091

165,348

74,997

283,305

27,682

62,773

* Unit costs and opportunity costs are equal for these excluded deliveries.

62

The Optimum

Solutions

for underground (U) and surface (S) deposits B.t.u.) district 10

11

12

13

29,180

123,221

14 47,005

11,708

Unused capacity 51,756 194,241 17,485 49,918 12,828

690 21,166

8,525

6,999 3,243

17,181 1,388 31,006

*

3,365 7,382 927 609 15,763 28,548 18,108

10,522

60,186

123,221 47,005 336,470

63

The Efficiency of the Coal Industry

Table 20. Minimum cost solution for 1949 (10 10 To district 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

U U

1

2

s

5

6

7

#

54,594

2,344 55,505

547 62,231 161,249

101,479 47,751

s s s s s s s s s

Demands

4

43,468

U

U U u u u u u u s

3

42,828 9,648 65,451

144,947

47,751

97,422

67,497

127,682

62,096

16,469 26,890

223,345

43,906

* Unit costs and opportunity costs are equal for these excluded deliveries.

64

The Optimum

Solutions

for underground (U) and surface (S) deposits B.t.u.) district10

11

5,847

12

13

19,074

90,878

14

Unused capacities

191,472 36,567 289,876 239,532 767

728

10,500 15,623

8,905

17,067 9,985

4,204 2,426 28,955

3,952

6,635

1,630

10,527 22,258 10,535

520 7,150

48,029

90,878

65

37,334 758,432

The Efficiency of the Coal Industry

Table 21. Minimum cost solution for 1951 (10 10 From district 1 2 3 4 5 6 7 8 9

U U U U U U U U U

10 u 11 u 1 s

2 3 4 5 6 7 8 9 10 11

S S S S S S S S S S

Demands

To 1 40,816

92,215

133,031

2 *

45,099

45,099

3

4

105,695

18,502 41,056

44,037

149,732

5

6

18,618 55,603

7 23,297

174,484

* 9,075

68,633

66,683

140,904

67,909

2,223 23,087

242,393

48,607

* Unit costs and opportunity costs are equal for these excluded deliveries.

66

The Optimum Solutions

for underground (U) and surface (S) deposits B.t.u.) district8

9

10

11

12 40,461

13

14

Unused capacities Capacities

135,114 102,105 42,901 222,760 158,138

7,159 419 6,510 12,908

7,695

5,025 3,744 2,263

9,820 11,832

*

4,568 8,599 1,372 290 11,497 21,507 11,192

6

7,985

52,293

*

175,930 426,845 305,632 41,056 55,603 174,484 7,159 419 32,138 13,564 2,263 92,215 56,931 44,037 9,075 66,683 70,132 23,087 4,568 8,599 1,372 290

102,105 42,901 534,203 1,612,082

67

The Efficiency of the Coal Industry in the appropriate solutions without changing their total costs. This lack of uniqueness results from an inability to distinguish between deliveries from the underground and surface deposits in certain districts. This situation can be illustrated by considering the deliveries from District 1 for 1947. The second and sixth columns of Table 22 contain the levels and costs of the deliveries from District 1 which are listed in Table 19. Now consider the effects of the introduction of a delivery x\h from the surface deposit in District 1 to District 5 which is excluded from the original optimum solution, but for which opportunity and unit costs are equal. The introduction of #f5 results in equal reductions of the levels of and x\x, and a corresponding increase in the level of The maximum level for x'jb is 48,419, at which point becomes zero. The levels and costs of deliveries for the new solution which is obtained by the introduction of iCjg 3X6 presented in the fourth and seventh columns of Table 22. The new solution is also a minimum cost solution which contains thirty-six deliveries and unused capacities with positive levels. The levels and costs of individual deliveries from District 1 are changed in the new solution, but the total costs of its deliveries and the outputs of its two deposits remain unchanged. Table 22. Changes in the minimum cost solution for 1947 resulting from the introduction of a delivery from the underground deposit in District 1 to District 5 Delivery Level

xi a 4

78.418 48.419 47,005

Change

+48,419 -48,419

173,842 «h x\6

142,982 0 142,982

New level

Unit cost (dollars)

Cost (dollars)

126,837 2,004 157,149,672 0 2,170 105,069,230 47,005 2,453 115,303,265

New cost (dollars) 254,181,348 0 115,303,265

173,842 -48,419 +48,419

94,563 1,329 190,023,078 48,419 1,495 0 142,982

68

567,545,245

125,674,227 72,386,405 567,545,245

The Optimum Solutions Deliveries from the underground and surface deposits in District 1 to District 5 have the same delivered price, which equals cost for the underground delivery ($2,170) and cost plus unit royalty ($1,495 + 675) for the surface delivery, and the consumer in District 5 is indifferent as to which satisfies his demand. An economic analysis cannot distinguish between alternatives with equal costs, but this type of indeterminacy is of no substantive importance. It results only if deliveries are included in an optimum solution from both of the deposits in a district, and unit transport costs are identical for coal from both deposits. Delivered prices would not be equal if unit transport costs differed for coal from the underground and surface deposits in a district. As a matter of convenience, the solutions given in Tables 19, 20, and 21 show the underground deliveries being sent to the more distant consumption locations, but an alternative scheme could be used if one was deemed desirable. The limited indeterminacy of the optimum solutions has no effect upon their total costs, or the outputs of the various deposits; these are unique.

STABILITY OF THE SHORT-RUN SOLUTIONS

The short-run solutions are based upon actual demands, capacities, and unit costs. The solutions for two years would be identical if these data remained unchanged. Changes in the levels of deliveries and unused capacities from one year to another reflect data changes. Large demand fluctuations during the period for which the model was applied are responsible for some rather large fluctuations in the levels of individual deliveries and unused capacities. However, most deposits maintained their market advantages, and there are relatively few shifts of deliveries and unused capacities into and out of the minimum cost solutions. Twenty-nine of the thirty-six included deliveries and unused capacities are included in the solutions for all three years. This is a remarkable degree of stability if it is remembered that the thirty-six are selected from a total of one hundred and ninety. The use of the short-run solutions to represent an efficient optimum would be highly questionable if the solutions failed to reveal this stability. A large number of year-to-year shifts from one delivery to another brought about by changing cost advantages 69

The Efficiency of the Coal Industry would suggest a divergence between a number of short-run solutions, and a solution for a longer period. The costs of numerous shifts would be too large to be ignored, and would have to be introduced explicitly in order to obtain an efficient solution for a longer period. Some of the small number of shifts which did take place were solely the result of demand changes. For example, the demand of District 6 for 1947 exceeded its combined capacities, and coal was imported from District 3. Its demand was less than its capacities in 1949 and 1951, and coal was exported to District 7. Shifts which result from demand changes can throw little doubt upon the validity of the short-run solutions. The stability of the short-run solutions is a result of the stability of relative unit costs. In fact this stability is so great that it suggests the possibility of short-range predictions of efficient solutions. If the unit costs for 1949 are used to determine an efficient solution together with the demands and capacities for 1951, the solution is the same as the one obtained for 1951 with only two exceptions. The underground delivery of 6,510 units from 9 to 8 is replaced by an underground delivery from 3 to 8, and 2,263 units of the underground delivery from 9 to 11 is replaced by an underground delivery from 11 to 11. COMPARATIVE STATICS

The method of comparative statics is used in economics to determine the effects of a given data change upon the variables of a system. The present model can be used to provide numerical answers to questions involving comparative statics. A simple example serves to illustrate both the application of comparative statics, and the interdependence of the variables of the present model. Consider the effects of a 50-percent increase in the demand of District 9 upon the efficient solution for 1951. These effects can be discovered by increasing the demand of District 9 from 21,507 to 32,261 units, and then recomputing the efficient solution for 1951. Table 23 contains the original and new values of the variables affected by this data change. The values of five deliveries, three unused capacities, three delivered prices, and 70

The Optimum Solutions Table 23. Effects of a 50-percent increase in the demand of District 9 upon the efficient solution for 1951 District

From 3 9 9 9 11

Method

to 8 8 9 11 11

Original

Change

Delivery levels (10 10 B.t.u.) U U U U U

— 6,159 12,246 7,673 —

Net change 3 9 11

New

2,431 3,728 23,000 5,410 2,263

+2,431 -2,431 +10,754 -2,263 +2,263 +10,754

U U U

Unused capacities (10 10 B.t.u.) 169,394 166,963 -2,431 6,060 — -6,060 2,263 — -2,263

Net change

-10,754

8 9 11

— — —

Delivered prices (current d o l l ys per 10 10 B.t.u.) 3,500 3,902 +402 2,498 2,900 +402 4,004 4,406 + 402

8 8 9 9 11 11

U S U S U S

Unit royalties (current dollars per 10 10 B.t.u.) 303 705 +402 1,212 1,614 +402 — 402 + 402 659 1,061 +402 —298* 104 + 402 1,414 1,816 +402

* The cost of a unit delivery from 11 to 11 exceeds the delivered price in District 11 by 298 dollars in the original efficient solution. six unit royalties are altered. The total of the deliveries is increased by 10,754 units, and the total of the unused capacities is decreased by a like amount. The increased demand in District 9 is met by an increase in the level of the delivery from the underground deposit in 9. The unused capacity of this deposit is reduced from 6,060 units to zero, and its deliveries to 8 and 11 are reduced by 2,431 and 2,263 units respectively. 71

The Efficiency of the Coal Industry Deliveries from the underground deposits in 3 and 11 are introduced to satisfy the demands in 8 and 11 respectively. All of the prices and royalties which are affected are increased by 402 dollars, the differential between the unit costs of a delivery from District 9 to District 8 and a delivery from 3 to 8. The largest cost increase brought about by a given data change permeates through the system. ERRORS OF OBSERVATION

If the functional relations of the model are assumed correct, deviations of actual quantities from the efficient are the result either of the nonconformity of the coal industry to the norm established by the model, or errors in the data. Errors of observation are present in nearly every numerical analysis, but are frequently ignored since there is no way to measure them. If " t r u e " figures were available for comparison, there would be no need to use estimated quantities. The question of the reliability of data is usually a matter of subjective confidence on the part of the analyst and others. 2 However, we can determine the effects of errors of observation upon the efficient solutions. Let us assume that a demand, capacity, or unit cost is in error by some arbitrary amount, and then determine how the efficient solution is affected when it is changed to its " t r u e " value. The comparative statics problem demonstrates that a single data change can affect the values of a number of variables. A new solution must be computed to determine the effects of each data change. In practice it is seldom necessary to perform a new computation since the effects of a single data change can usually be determined by inspection, particularly if the change is not large. An error of observation for a demand affects only one of the efficient deliveries if the deposit which makes the highest cost delivery satisfying that demand has a sufficient margin of unused capacity. In the typical case, an increase (decrease) of a demand by not more than 5 percent would result in an increase (decrease) 2 A detailed description of the data and rectification methods used for the implementation of the model is contained in Appendix A. The reader can evaluate them, and determine his own level of subjective confidence.

72

The Optimum Solutions of the highest cost delivery which satisfies that demand, and a decrease (increase) of its unused capacity by the amount of the error. For example consider the demand for District 8 for 1947. If it were overestimated by 5 percent (788 units), the delivery from the underground deposit in District 3 to District 8 would be overstated, and its unused capacity understated by 788 units. The above situation holds for all of the demands except for those of District 2 for all three years and District 6 for 1949 and 1951. Errors in the demands for Districts 2 and 6 would also affect the delivery levels to Districts 12 and 7 respectively. If the demand for District 2 for 1947 were overestimated by 5 percent (3,314 units), the delivery from the surface deposit in District 2 to District 2 would be overstated, the delivery from this deposit to District 12 understated, the delivery from the underground deposit in District 2 to District 12 overstated, and the unused capacity understated by 3,314 units. In either situation, reasonable errors of observation for the demands cannot exert a large influence upon the efficient solutions. Errors of observation of not more than 5 percent for the capacities for which the efficient solutions do not dictate full utilization have no effect upon the efficient sets of deliveries. If a capacity is overestimated (underestimated), its unused capacity level is overstated (understated) by the same amount. The capacity of the underground deposit in District 1 for 1947 provides an example. If it is overestimated by 5 percent (11,278 units), the only error is that its unused capacity is stated at 51,756 rather than 40,478 units. Delivery levels would be affected only if the capacity of one of these deposits was underestimated by an amount greater than the level of its unused capacity. Errors of observation for the capacities of deposits for which the efficient solutions dictate full utilization would affect two or more of the delivery levels. An error usually results in the overstatement of one delivery level and the understatement of another by the amount of the error. If the capacity of the surface deposit in District 5 were overestimated by 5 percent (2,617 units) for 1947, the delivery from that deposit to District 5 would be overstated, the delivery from the underground deposit in District 1 to District 5 understated, and the unused capacity of this deposit overstated by 2,617 units. In a few cases—for example, the 73

The Efficiency of the Coal Industry surface deposit in District 2—an error would affect more than two delivery levels. Errors of observation for the unit costs can have either no effect upon the set of efficient delivery levels or a greatly magnified one. An efficient solution is a set of deliveries which minimizes total cost, and the unit cost of a delivery which is included in an efficient solution is less than the net cost of any delivery or combination of deliveries which could be substituted for it. Let us imagine a situation in which the unit cost of an included delivery is increased, and the other data remain unchanged. Eventually the unit cost of this delivery will reach a critical level at which it equals the net cost of its least costly alternative. Some of the differentials between the unit costs of the included deliveries and their least costly alternatives may be very large; others may be very small. The delivery from the underground deposit in District 1 to District 14, which has a level of 47,005 units for 1947, provides an example of a delivery with a narrow cost differential. Its unit cost for 1947 is 2,453 dollars. Its least costly alternative (a delivery from the underground deposit in District 2) has a unit cost of 2,463 dollars. If the cost of the delivery from District 1 were eleven dollars greater, or the cost of the delivery from District 2 eleven dollars less, the entire delivery would be assigned to District 2. The cost differential in this case is very narrow, only ten dollars. Cost differentials for included deliveries can be determined through an examination of the unit and opportunity costs for deliveries excluded from the minimum cost solution. The opportunity cost of an excluded delivery is the net reduction in the costs of the included deliveries which would result from its introduction at the unit level. If the unit cost of an excluded delivery is less than its opportunity cost (ci} .

m^*r^u">co »O T-H rH

00 © m ^O n i f l o N i o 00" cn" ^ CS IT)

n "i So o o i o o o v u i ^ i ^ t ^ m 11i CO N n

CO

T-h r-» i f o\

OmrtrtNiflOtSMnt^iOrt COCOvOOcOCOrHUlrtC^ f S

•gvOCOCN o

cor^

m T u-TaT •F I c f co T t - o TJm (N oov>ooorsiooavoom M>CM\oa!xO«!rHTHOvmmin o o n i f l i n M i f i n N O i n ^ N n cs TJ-'co f ^ OO" © vO OOOvOT-

See above, pp. 26-27. George B. Dantzig, "Maximization of a Linear Function of Variables Subject to Linear Inequalities," Activity Analysis, pp. 339-347. 1

137

A SELECTED BIBLIOGRAPHY BOOKS Adams, Walter (ed.). The Structure of American Industry. Rev. ed., New York, 1954. Ayres, Eugene, and Charles A. Scarlott. Energy Sources: The Wealth of the World. New York, 1952. Baker, Ralph Hillis. The National Bituminous Coal Commission. Baltimore, 1941. Baratz, Morton S. The Union and the Coal Industry. New Haven, 1955. Barger, Harold, and Sam H. Schurr. The Mining Industries, 18991939. New York, 1944. Chamberlin, Edward H. The Theory of Monopolistic Competition. 7th ed., Cambridge, Mass., 1956. Charnes, A., W. W. Cooper, and A. Henderson. An Introduction to Linear Programming. New York, 1953. Dean, William, H., Jr. The Theory of the Geographical Location of Economic Activities. Ann Arbor, 1938. Dorfman, Robert. Application of Linear Programming to the Theory of the Firm. Berkeley, 1951. Dron, Robert W. The Economics of Coal Mining, London, 1928. Evanson, Howard N. The First Century and a Quarter of American Coal Industry. Pittsburgh, 1942. Fisher, Waldo E., and Charles M. James. Minimum Price Fixing in the Bituminous Coal Industry. Princeton, 1955. Fritz, Wilbert G., and Theodore A. Veenstra. Regional Shifts in the Bituminous Coal Industry. Pittsburgh, 1935. Glover, J. G., and W. B. Cornell (eds.). The Development of American Industries. 3rd ed., New York, 1951. Hesse, A. W. The Principles of Coal Property Valuation. New York, 1930. Hicks, J. R, Value and Capital. 2nd ed., Oxford, 1946. Hoover, Edgar M. The Location of Economic Activity. New York, 1948. 139

A Selected Bibliography Koopmans, Tjalling C. (ed.). Activity Analysis of Production and Allocation. New York, 1951. Lerner, A. P. The Economics of Control. New York, 1944. Little, I. M. D. The Price of Fuel. Oxford, 1953. Moore, Elwood S. Coal: Its Properties, Analysis, Classificationt Geology, Extraction, Uses and Distribution. 2nd ed., New York, 1940. Parker, Glen L. The Coal Industry: A Study in Social Control. Washington, D.C., 1940. Parsons, A. B. (ed.). Seventy-five Years of Progress in the Mineral Industry, 1871-1946. New York, 1947. Robinson, Joan. The Economics of Imperfect Competition. London, 1933. Samuelson, Paul A. Foundations of Economic Analysis. Cambridge, Mass., 1947. Shurick, A. T. Coal Mining Costs. New York, 1922. Smart, R. C. The Economics of the Coal Industry. London, 1930.

REPORTS AND OTHER SPECIAL PUBLICATIONS

v

Association of American Railroads, Railroad Committee for the Study of Transportation. Bituminous Coal and Lignite. March 1946. . Report on Anthracite and Bituminous Coal. June 1947. Anglo-American Council on Productivity. Coal. Report of a Productivity Team Representing the British Coal Mining Industry which Visited the United States of America in 1951. London, December 1951. Backman, Jules. Bituminous Coal Wages, Profits and Productivity. Prepared for the Southern Coal Producers' Association, and presented before the Presidential Coal Board. Washington, D.C., February 1950. Barnett, Harold J. Energy Uses and Supplies 1939,1947,1965. U.S., Bureau of Mines, "Information Circular," No. 7582. Washington, D.C., October 1950. Beckmann, M., and T. Marschak. On the Theory of Location in the Short-run. Stanford University, Department of Economics, "Technical Reports," No. 11. 26 March 1954. Bituminous Coal Institute. Bituminous Coal Annual. Washington, D.C., published annually. Cowan, Donald R. G. More Capital Equipment: Coal's Foremost Economic Need. Washington, D.C. (1948?).

140

A Selected Bibliography Dantzig, G., and A. Orden. Notes on Linear Programming. Part II, "Duality Theorems." RAND Corporation. Santa Monica, 10 April 1953. Fieldner, A. C., W. E. Rice, and H. E. Mason. Typical Analyses of Coals of the United States. U.S., Bureau of Mines, "Bulletin," No. 446. Washington, D.C., 1942. Flynn, George J., Jr. Average Heating Values of American Coals by Rank and by States. U.S., Bureau of Mines, "Information Circular," No. 7538. Washington, D.C., December 1949. Graham, Herman D. The Economics of Strip Coal Mining. University of Illinois, Bureau of Economic and Business Research, " Bulletin," No. 66. Urbana, 1948. Houthakker, H. S. On the Numerical Solution of the Transportation Problem. Stanford University, Department of Economics, " Technical Report," No. 15. 23 December 1954. Hultgren, Thor. Freight Rates on Bituminous Coal with Index Numbers, 1929-1940. U.S., Interstate Commerce Commission, Bureau of Transport Economics and Statistics, Statement No. 413, File No. 26 C-7. Washington, D.C., February 1941. James, Charles M. Measuring Productivity in Coal Mining. University of Pennsylvania, Wharton School of Finance and Commerce, Industrial Research Department, "Research Report," No. 13. Philadelphia, March 1952. Kiessling, O. E., F. G. Tryon, and L. Mann. The Economics of Strip Coal Mining. U.S., Bureau of Mines, " Economic Paper," No. 11. Washington, D.C., 1931. Lake Carriers' Association. Annual Report. Cleveland, published annually. McCloud, Leland W. Comparative Costs of Competitive Fuels. West Virginia University, "Business and Economic Studies," Vol. I, No. 4. June 1951. National Association of Manufacturers. The Economic Impact of an Industry-Wide Strike: A Case Study of the 1949-1950 Coal Strike. "Economic Policy Division Series," No. 27. New York, July 1950. Ore and Coal Exchange. Lake Coal: Statement of Bituminous Coal Loaded into Vessels. Cleveland, published monthly. United Nations. Proceedings of the United Nations Scientific Conference on the Conservation and Utilization of Resources. Vol. Ill, Fuel and Energy Resources. New York, 1951. United States, Bureau of Mines. Minerals Yearbook. Washington, D.C., published annually.

141

A Selected Bibliography United States, Interstate Commerce Commission, Bureau of Transport Economics and Statistics. Carload Waybill Analyses: Stateto-State Distribution of Bituminous Coal Traffic and Revenue. Washington, D.C., published annually. , Interstate Commerce Commission, Bureau of Transport Economics and Statistics. Comparative Statement of Railway Operating Statistics: Individual Class I Steam Railways in the United States. Washington, D.C., published annually. , President's Materials Policy Commission. Resources for Freedom. Vol. Ill, The Outlook for Energy Sources. Washington, D.C., June 1952. , Office of Temporary Controls, Office of Price Administration, Economic Data Analysis Branch. Survey of Commercial Bituminous Coal Mines. "OPA Economic Data Series," No. IS. Washington, D.C., n.d. , Transportation Investigation and Research Board. The Economics of Coal Traffic Flow. Senate, 79th Congress, 1st Session, Document No. 82. Washington, D.C., 1945. University of Maryland, Bureau of Business and Economic Research. Coal in the Maryland Economy, 1736 to 1965. "Studies in Business and Economics," Vol. 7, No. 3. December 1953.

ARTICLES Adelman, M. A. " T h e Measurement of Industrial Concentration", Review of Economics and Statistics, XXXIII (November 1951), 269-296. Bain, Joe S. "The Profit Rate as a Measure of Monopoly Power," Quarterly Journal of Economics, LV (February 1941), 271-293. Carlisle, Donald. " The Economics of a Fund Resource with Particular Reference to Mining," American Economic Review, XLIV (September 1954), 595-616. Charnes, A., and W. W. Cooper. " T h e Stepping Stone Method of Explaining Linear Programming Calculations in Transportation Problems," Management Science, I (October 1954), 49-70. , W. W. Cooper, and B. Mellon. " Blending Aviation Gasolines : a Study in Programming Interdependent Activities in an Integrated Oil Company," Econometrica, XX (April 1952), 135-159. , W. W. Cooper, and B. Mellon. "A Model for Programming and Sensitivity Analysis in an Integrated Oil Company," Econometrica, XXII (April 1954), 193-217. 142

A Selected

Bibliography

Chipman, John. "Linear Programming," Review of Economics and Statistics, XXXV (May 1953), 101-117. . "Computational Problems in Linear Programming," Review of Economics and Statistics, XXXV (November 1953), 342-3+9. Christenson, C. L. " T h e Theory of the Offset Factor: The Impact of Labor Disputes upon Coal Production," American Economic Review, XLIII (September 1953), 513-547. . " T h e Impact of Labor Disputes upon Coal Consumption," American Economic Review, XLV (March 1955), 79-112. Crain, Harry M. (ed.). "Economics of the Mineral Industry," Quarterly of the Colorado School of Mines, XLV (January 1950), 1-47. Dorfman, Robert. " Mathematical or ' Linear' Programming," American Economic Review, XLIII (December 1953), 797-825. Dunlop, John T. "Price Flexibility and the ' Degree of Monopoly,' " Quarterly Journal of Economics, LIII (August 1939), 522-534. Fishman, Leon, and Betty G. Fishman. " Bituminous Coal Production during World War II," Southern Economic Journal, XVIII (January 1952), 391-396. Gray, L. C. "Rent under the Assumption of Exhaustibility," Quarterly Journal of Economics, XXVIII (May 1914), 466-^89. Henderson, James M. " A Short-run Model for the Coal Industry," Review of Economics and Statistics, XXXVII (November 1955), 336-346. . "Efficiency and Pricing in the Coal Industry," Review of Economics and Statistics, XXXVIII (February 1956), 50-60. Hitchcock, Frank L. " T h e Distribution of a Product from Several Sources to Numerous Localities," Journal of Mathematics and Physics, Massachusetts Institute of Technology, XX (1941), 224-230. Hotelling, Harold. " T h e Economics of Exhaustible Resources," Journal of Political Economy, XXXIX (April 1931), 137-175. Isard, Walter. " T h e General Theory of Location and SpaceEconomy," Quarterly Journal of Economics, LXIII (November 1949), 476-506. Koopmans, Tjalling C. "Optimum Utilization of the Transportation System," Econometrica, XVII (July 1949, supplement), 136-145. . "Efficient Allocation of Resources," Econometrica, XIX (October 1951), 455-^65. Lang, R. T. " Bituminous Stripping Future in Pennsylvania," Coal Age, LV (January 1950), 68-69.

143

A Selected Bibliography Lerner, A. P. "The Concept of Monopoly and the Measurement of Monopoly Power," Review of Economic Studies, I (June 1934), 157-175. Morgan, Theodore. "A Measure of Monopoly in Selling," Quarterly Journal of Economics, LX (May 1946), 461-463. Orchard, John E. " T h e Rent of Mineral Lands," Quarterly Journal of Economics, XXXVI (February 1922), 290-318. Papandreou, A. G. "Market Structure and Monopoly Power," American Economic Review, XXXIX (September 1949), 883-897. Rothschild, K. W. " T h e Degree of Monopoly," Econometrica, IX (February 1942), 24-39. Samuelson, Paul A. "Spatial Price Equilibrium and Linear Programming," American Economic Review, XLII (June 1952), 283-303. Smith, R. Tynes, III. "Technical Aspects of Transportation Flow Data," Journal of the American Statistical Association, IL (June 1954), 227-239. Somers, Gerald G. "Effects of North-South Wage Uniformity on Southern Coal Production," Southern Economic Journal, XX (October 1953), 121-129.

144

Index Adelman, M. A., 15 Agricultural land, classification of, 133— 134 Aitken, A. C., 23n American Institute of Mining and Metallurgical Engineers, 45 Bain, J. S., 5 Baratz, M . S., 4n Capacities: for agriculture, 134—135; conversion to heating values, 132; definition of, 45-46; estimation by 280-day rule, 45-46, 122-123; location of, 47-48; possible reductions of, 99-102; of surface deposits, 46—47, 96; unused, 92-93 Captive mines, 111 Chamberlin, E. H., 3 Charnes, A., 6n, 26n, 29n, 31n, 32n Christenson, C. L., 109n Coal industry: compared with agriculture, 99, 116-117; growth of, 93-95; labor force, 100; nationalization of, 102-103; organization of, 98-99; output, cyclical variation of, 41, 106-109; profitability of, 91 Coal mines, number of, 103-105 Coal reserves, 14 Coals: classification of, 41-42; output index for, 4 2 ^ 3 , 131 Coking coals, 42, 122 Comparative statics, 70-72 Competition, nonperfect, theories of, 2-3. See also Monopoly Competition, perfect: necessary conditions for, 1-2, 5-6; results of, 2; results compared with dual system, 25, 87-91; and welfare norms, 3-6 Consumption of coal: classification of, 15, 43; in the future, 101-102. See also Demand for coal Cooper, W. W., 6n, 26n, 29n, 31n, 32n Cost differentials, 74-77 Costs, total, 8-9, 20-21, 85-86. See also Extraction costs; Opportunity costs; Transport costs Cowan, D. R. G., 116n

Dantzig, G . B., 7n, 26n, 29n, 32n, 35n, 139n Data-gathering units, 119 Degeneracy, 31, 61 Demand for coal: cyclical fluctuations of, 106; estimation of, 44-45, 120-122; treatment of, 11-12, 15-16. See also Consumption of coal Demand curves, 11-12 Deposits, classification of, 14—15, 43, 54— 55 Distribution, marginal productivity theory of, 2 Dorfman, R., 6n Dual system: for agriculture, 136-137 description of, 7-8, 25-28; and distribution, 87-91; and errors of observation, 79-80; method of solution, 38-40; solutions of, 77—79 Duality theorems, 26-27 Dunlop, J. T „ 5n Efficiency, 11-12, 21, 81, 85. See also Inefficiency Efficiency index, 83-85, 92-93 Efficient solutions. See Solutions Errors of observation, 72-77 Evanson, H. N., 97n Exports of coal, 15, 122 Extraction costs, 8, 16-19, 48-50, 55-60, 124-126 Extraction functions, 13, 16-17 Fieldner, A. C„ 41 n Fisher, W . E„ 42n, 103n Fishman, L. and B. G., lOln Flynn, G. J., Jr., 129n Fritz, W. G., 103n Gale, D., 26n Graham, H. D., 49n Heating value units, 42-43, 129, 132 Henderson, A., 6n, 26n, 3 In Hicks, J. R., In Hitchcock, F. L., 7n Hoel, P. G „ 55n Hotelling, H., 14n Houthakker, H . S„ 32n

145

Index Imports of coal, 15, 120 Indeterminacy, 37-38, 61, 68-69 Inefficiency: causes of, 92-96, 108-109; costs of, 86-91, 92, 114; reductions of, 99-100, 103-105; union imposed, 109112. See also Efficiency Interior Department, Bituminous Coal Division, 103 James, C. M „ 42n, 103n Kiessling, O. E., 49n Koopmans, T . C., 6n, 7 Kuhn, H. W., 26n Lerner, A. P., 3, 5 Linear programming: application to agriculture, 136; application to the coal industry, 6-12; format, 21-25; a method of solution for, 28—40. See also Dual system; Solutions Little, I. M . D., 102n Mann, L., 49n Marginal cost: of coal, 10; pricing, 2, 5 Marshall, A., 18n Maryland, output adjustments in, 98 Mason, H. E., 41n Mills, F. C., 5On Minimum cost solutions. See Solutions Mitchell, W . C., 106n Monopoly, 3, 5, 21 Monopoly power, measures of, 5 Morgan, T., 6n National Bituminous Coal Commission, 103 National Coal Board, 102-103 Opportunity costs: of excluded deliveries, 34—35; relation to total cost, 37; of u n used capacities, 38-40 Optimum solutions. See Solutions Orchard, J. E., 18n Orden, A., 26n Output index. See Coals, output index for Paley Commission, 101 Papandreaou, A. G., 6n Parker, G . L., 4n Perfect competition. See perfect Perlis, S., 23n

Competition,

President's Materials Policy Commission, lOln Prices: actual, 87-89; average weighted, 89-90; competitive, 26-27, 87-89; reductions of, 91; setting of, 102-103 Railroads: coal carried by, 50; coal consumed by, 120-122; transport costs for coal, 126-129 Reiter, S., 7n Rice, W. E., 41 n Robinson, J., 3, 4 Rossander, A. C., 55n Rothschild, K. W., 6n Royalties: effects of observation errors on, 79-80; normal contrasted with differential, 18; payments of, 90-91; relation to capacity utilization, 27; relation to prices and costs, 11, 77; solution values for, 78-79 Samuelson, P. A., I n Schmooker, J., 4n, 109n Short-run models, applications of, 116— 117 Short-run period, 7, 13, 41 Smith, R. T . , I l l , 126n Solutions: basic feasible, 30-34; computation methods, 28-40; effects of errors of observation on, 72-77; feasible, 9, 20, 30; as a guide for capacity reductions, 100; and long-run changes, 97-98; nature of, 7-8, 11, 20-21, 28, 85; and output adjustments, 106-107; stability of, 69-70; uniqueness of, 61, 68-69 Somers, G. G., 87n Technological matrix, 23, 28-29, 136 Tests of the model, 54-60 Tidewater shipments of coal, 128 Transport costs, 19-20, 50-51, 126-129, 135 Truck, coal shipments by, 127 Tryon, F. G., 49n Tucker, A. W., 26n United Mine Workers of America, 109112 United States Coal Commission, 103 Variables of the model, 7, 15, 25-26, 61, 133 Variance ratios, 56-58 Veenstra, T . A., 103n

146