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A Constrained Price Discriminator: An Application To The U.S. Post Office

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A Constrained Price Discriminator: An Application To The U.S. Post Office

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Content j 72-556 } \ GILON, Paul Raphael, 1933- ! A CONSTRAINED PRICE DISCRIMINATOR: AN APPLICATION TO THE U.S. POST OFFICE. University of Southern California, Ph.D., 1971 Economics, general University Microfilms, A X ERO X Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED A CONSTRAINED PRICE DISCRIMINATOR: AN APPLICATION TO THE U.S. POST OFFICE by Paul Raphael Gilon A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY •(Economics) June 1971 UNIV ERSITY OF SO U TH ER N CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFO RNIA 9 0 0 0 7 This dissertation, written by PAUL RAPHAEL GILON under the direction of / i . i . ? . . . . . Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments of the degree of D O C T O R OF P H IL O S O P H Y Dean Date J-ili DISSERTATION COMMITTEE Chairman DEDICATION This work is to be dedicated first to the memory of my parents Solomon and Dwora Gutblatt. Secondly, to my wife Sue and my little girls Diana and Simone. Thirdly, to Doctor Le Mehaute and Doctor Sam Fersht for having in spired me in seeking knowledge. My sincere thanks are also extended to the library staff at Cal State Long Beach, who on a state college budget render a university level service. Notably, Mr. Roman Kochin and Mrs. Linda Steel were of great help. In addition, this dissertation could not have been accomplished without the assistance of the dedicated computer staff at Cal State Long Beach. Specifically, I am grateful to Mr. Frank Brennan, Mr. Ralph Frisbie, Mr. Bruce Hanks, and to Miss Lynn Lewis for their support. The whole staff should further be con gratulated for running a college computer laboratory on a high school budget. Last but not least, I extend my thanks to Emily Lowe for typing from a rather rudimentary cuneiform without the assistance of a Rosetta Stone. ii TABLE OP CONTENTS Page LIST OP TABLES.......... vi LIST OP ILLUSTRATIONS............................... viii Chapter I. INTRODUCTION .............................. 1 Scope........................................ 1 Method ........................................ 2 II. REVIEW OP THE LITERATURE...................... 5 Introduction................................... 5 Dupuit ........................................ 7 Marshall ...................................... 9 P i g o u.......................................... 15 Robinson........................................ 21 Leontief........................................ 28 Clemens........................................ 32 III. THEORETICAL STRUCTURE ....................... Ml Theoretical Models ........................... Ml The Models iii i iv Chapter Page Equilibrium Considerations Constrained Versus Unconstrained Equilibrium Algebraic Models ........................... . 51 The Models Regular demand Discount demand Average cost function Equilibrium Prices and Output Allocation of Costs Constrained versus Unconstrained Prices Summary ................................... . 64 IV. EMPIRICAL WORK The Empirical Firm ......................... • 65 The Data ................................... . 66 Data Problems Significant Figures Consolidation of Classes II, III, and IV into the Discount Market Empirical Models ........................... . 76 Class I Demand Average Cost Models Demand for Discounted Mail Evaluation and Summary V Chapter Page V. APPLICATIONS....................................99 Selection of the Pinal Models............... 100 Regular Demand Discount Market Demand Average Cost Function Summary Simulation.....................................108 Mechanization Results of the Simulation Remarks Findings....................................... 122 The Discount Rate d' Controlling Monopoly Income Varies, d Constant Income Constant, d Varies VI. CONCLUSION.....................................129 Summary....................................... 129 Evaluation.....................................130 Recommendations for Further Research ........ 132 APPENDIX............................................. 134 BIBLIOGRAPHY ........................................ 155 LIST OP TABLES Table Page 1. Hypothetical Tea Consumer........................ 12 2. Total Revenue and Total Cost of Mail by Class . 67 3. 19^5 Method Ratios.............................. 70 4. Gross National Product or Expenditure 1929-1966 .................................... 73 5. Class I Demand Nomenclature .................... 79 6. Average Cost Models Nomenclature ............. 86 7. Discount Demand Nomenclature ................. 90 8. Discount Demand Parameters .................... Ill 9. Disposable Income = 600 x 10^ Dollars...........113 10. Disposable Income = 650 x 10^ Dollars ......... 115 11. Disposable Income = 700 x 10^ Dollars ......... 117 12. Discount to the Regular Market Due to the Constraint: d’ = ^23 13. Disposable Income = 100 x 10^ Dollars ......... 136 14. Disposable Income = 150 x 10^ Dollars ......... 138 15. Disposable Income = 200 x 10^ Dollars ......... 140 16. Disposable Income = 250 x 10^ Dollars...........142 17. Disposable Income = 300 x 10^ Dollars ......... 144 vi Table 18. Disposable Income = 350 19. Disposable Income = 400 20. Disposable Income = 450 21. Disposable Income = 500 22. Disposable Income = 550 vli | Page x 10^ Dollars......... 146 x 10^ Dollars......... 148 x 10^ Dollars......... 150 x 10^.Dollars......... 152 x 10^.Dollars......... 154 LIST OP ILLUSTRATIONS Figure Page 1. Dupuit's Tax M o d e l ........................... 8 2. Demand for T e a ..................................11 3. Price Discrimination ......................... 20 4. Equilibrium Output and Prices ................. 24 5. Output with Price Discrimination...............26 6. Multiple Product Pricing and Output ........... 36 7- Average Revenue and Realized Average Revenue . 39 8. Market Creation..................................44 9. Double-Edged Weapon ........................... 53 10. Various Normative Prices .................... 63 11. The Empirical Models.......................... 107 12. Disposable Income = 600 x 10^ Dollars.......... 112 13. Disposable Income = 650 x 10^ Dollars.......... 114 14. Disposable Income = 700 x 10^ Dollars.......... 116 15* Real Disposable Income x 1 0 ^ ..................124 16. Profits versus Discount ....................... 126 17. d Versus d ' ................................... 127 viii Figure Page 18. Disposable Income = 100 X C T \ o I —1 Dollars . • • . 135 19. Disposable Income = 150 X 109 Dollars . . 137 20. Disposable Income = 200 X 109 Dollars . • 139 21. Disposable Income = 250 X 109 Dollars . . 141 22. Disposable Income = 300 X 109 Dollars . . 143 23. Disposable Income = 350 X 109 Dollars . . 145 24. Disposable Income = 400 X 109 Dollars . . 147 in C\J Disposable Income = 450 X 109 Dollars . . 149 26. Disposable Income = 500 X 109 Dollars . . 151 27- Disposable Income = 550 X 109 Dollars . . 153 CHAPTER I INTRODUCTION Scope The objective of this dissertation is to analyze the behavior of a third degree'*' price discriminating monopolist operating in two markets under the constraint, P1 = (1-d) ?2 where P^ and P2 are the prices and d is a given discount rate. The value for d may be set by contract or policy. An example of the latter is the practice by many department stores to discount merchandise sold to employees by 10 percent. Specifically, we wish to examine the effect of the discount rate on the equilibrium values, P2 and Q2. That is to say, how do the consumers in the second mar ket "pay" for the discount benefitting the consumers of market one. ^In the remainder of this paper, with the excep tion of Chapter II (the review) we refer to "third degree price discrimination" as "price discrimination." Theoretically, the general linear constraint is: P1 = (1 - d) P2 - a where a is also a given policy parameter. The case for d = 1 and a ^ 0 was solved for a 2 profit maximizer by Leontief. This dissertation addresses itself to the case where a = 0 and 0 < d _< 1. The two cases are not equivalent for if we eliminate from P - j ^ = P2 - a, and P.^ = (1 - d) P2, we obtain d = (a/P2). Clearly, d depends on a and ?0. I The analysis covers the long run. It was felt j | that the short run behavior of discounting was more a I problem in marketing. Furthermore, the data itself spans the years, 1933-1969. Method In order to measure the effect of d in market number two, we proposed the following impact measure. Let Pr represent the equilibrium constrained price. Let P^ represent the unconstrained price. Then, a measure of impact d' due to d is: 2W. W. Leontief, "The Theory of Limited and Un limited Discrimination," Quarterly Journal of Economics, LIV (1939-1940), 490-501. Leontief gave the name of unlimited price discrimination for a ^ 0, and limited price discrimination for a = 0. 3 d' = PA " Pr A •3 Graphically, we have: induced discount rate 0 d policy discount rate The investigation, and hence, the organization of this paper proceeds as follows: Chapter I (the present one), covers the scope and method of this project. Chapter II presents a his torical sketch of the problem. In Chapter III, we present ■ 3 The shape and slope of this curve depend on the demand functions and on the unconstraining method. Hence, the graph presented here applies specifically to the em pirical models fitted in this dissertation and for the unconstrained method just presented. The exact curve is presented in Chapter V. the theoretical models for our two markets. Chapter IV j 2 i presents the empirical fit over these theoretical models. In Chapter V, we apply the theoretical coefficients of Chapter IV to the theoretical models of Chapter III. Because of the parametric nature of the solution, we re sorted to simulation techniques. This is done in Chapter V. Finally, Chapter VI concludes this dissertation, with a brief evaluation, summary and recommendations for further research. l i A futile attempt was made to obtain business data. In light of public and congressional concern over the financial state of the Post Office, e.g., J. George Butler, "Toward Postal Reorganization," Christian Century, LXXXVII, No. 4 (January 28, 1970), 104-108; "A National Humiliation," editorial, Nation, CCX, No. 13 (April 6, 1970), 386-388; "Federal Workers March to a New Drummer," Business Week, MMCXVII (March 28, 1970), 40-42; "How a Post Office 'Corporation1 Would Work," Changing Times, XXIV, No. 2 (February, 1970), 17-20; and David Sanford, "Post Office Blues," New Republic, CLXII, No. 12 (March 12, 1970), 19-22, and in light of the data, we considered the following hypothetical case: what if the Post Office were to become a profit maximizer price discriminator? (A more detailed explanation is presented in Chapter IV in the section entitled "The Empirical Firm.") CHAPTER II REVIEW OP THE LITERATURE Introduction Schumpeter1 traces the evolution of price dis crimination from Dupuit through Pigou. He then notes that both Cournot and Marshall ”... failed to give ade quate attention to one very important aspect of monopo- 2 list strategy, Price Discrimination." ■ 3 Edgeworth also attributes the theory of price discrimination to Dupuit. The following statement bears this out. The feature of that theory with which we are now concerned is the power of the monopolist to discriminate between dif ferent species of commodities and customers, not preserving that unity of price which characterizes a perfectly competitive 1Joseph A. Schumpeter, History of Economic Analysis (New York: Oxford University Press, 195*0, p. 978. 2Ibid., p. 978. ^F. Y. Edgeworth, Papers Relating to Political Economy (London: MacMillan and Co., LtdT, 19^5.) • 6 market. The subject may fittingly be in troduced by a quotation from the earliest and still, I think, the highest authority on the theory of discrimination, Dupuit. ^ In a later passage, Schumpeter"^ further designates Dupuit as the discoverer of consumer surplus and Marshall as the rediscoverer some fifty years later. In light of these considerations and because of the great gaps in the successive occurrences of the con tributions in price discrimination, the review of the literature will follow in a chronological order. Thus, the first author to be reviewed is Dupuit. This is fol lowed by a brief summary of Marshall's consumer surplus. Thirdly, we turn to Pigou, who formalized the theory of price discrimination and who also introduced the degree- classification scheme. The fourth author in the review is Robinson. Her contribution is the rigorous analysis of price discrimination in terms of ijiarginal analysis. Next in the review is Professor Leontief for his contri bution of limited and unlimited price discrimination. Finally, the last author is Professor Clemens, who by an ingenious method, reconciled the pricing of multiple products and price discrimination. Under certain 1 1 Ibid. , p. 404. ^Schumpeter, p. 1070. ! assumptions, he showed them to be identical. Dupuit In 1844, Dupuit, a French railway engineer, pro posed a model to compute the loss of utility resulting g from the imposition of a per unit tax. He termed the area under the utility curve above the price "utilite 7 relative." Since he did not distinguish between demand and utility, this area corresponds to Marshall's "con- O sumer surplus," and the loss of utility due to the tax is equal to the change in "utilite relative." Figure 1 gives the model. From this model, Dupuit derived rules indicating how the tolls for public services should be levied. Of greater importance to this paper, however, is Dupuit's further reasoning that more revenues could be raised by grouping consumers according to the utility derived for Q equal services. c I. H. Rima, The Development of Economic Analysis (Homewood, Illinois: Richard D. Irwin, 1967), P* 184. 7Ibid., p. 184. Q Mark Blaug, Economic Theory in Retrospect (Home wood, Illinois: R. D~ Irwin, 1968), p. 360. ^Rima, p. 186. 8 Figure 1 DUPUIT'S TAX MODEL Price tax yield utilite relative Net loss of utility quantity SOURCE: Rima, p. 185. 9 How close this came to present day thought10 can be observed if we regroup consumers according to equal 11 elasticities instead of equal utilities. Marshall 12 Though Marshall did not address himself directly to discrimination, he did provide a fundamental tool for its analysis, e.g. for the impact of its existence, con- | sumer surplus. For this reason, Marshall is reviewed along with Pigou, Robinson, and other theorists in price discrimination. 13 The use of consumer surplus in its rediscover's own words is: Our aim now is to apply the notion of consumers’ surplus as an aid in es timating roughly some of the benefits which a person derives from his environ ment or his conjuncture. In order to give definiteness to our notions, let us consider the case of tea purchased for domestic consumption. Let us take the case of a man ... 10Though strictly speaking, we would have a case in quantity discrimination. 11Rima, p. 189 and Blaug, p. 361. ^Schumpeter, p. 978. 13 The discoverer being Dupuit. Schumpeter, p. 1070. 1 10 who, If the price of tea were 20s. a pound, would just be induced to buy one pound annually; who would just be induced to buy two pounds if the price were 14s., three pounds if the price were 10s., four pounds if the price were 6s., five pounds if the price were 4s., six pounds if the price were 3s., and who, the price being actually 2s., does purchase seven pounds. We have to investi gate the consumers' surplus which he derives from his power of purchasing tea at 2s. a pound. As an example, Marshall provides the schedule depicted in Figure 2 for the annual demand for tea by a hypothetical consumer. For this conjectured demand schedule, Marshall arbitrarily selects 2 shillings (hence 7 pounds) as the equilibrium price. With this simple model, Marshall con structs the concept of consumer surplus, which is briefly described below. For exposition purposes, Table 1 has 15 been reconstructed from Marshall’s verbal description. Columns (1) and (2) represent the schedule. Column (3) equals column (1) times column (2). Column (4) is best described by Marshall as follows: The fact that he would just be in duced to purchase one pound if the price were 20s., proves that the total enjoyment 14 Alfred Marshall, Source Readings in Economic Thought, ed. Philip C. Newman^e^aTT^ tFT^TTorkl W. f ? . Norton & Company, Inc., 1954), p. 434. 15Ibid., pp. 434-435. Price i n Shillings 11 Figure 2 DEMAND FOR TEA P 20 • 18 - 10 change in consumer surplus 3*15 6 7 8 1 2 Quantity Demanded (lbs) Table 1 HYPOTHETICAL TEA CONSUMER Price (1) Quantity (2) Total Expenditures (3) Minimum Worth To Buyer (4) Consumer Surplus (5) Extra Consumer Surplus (6) Cumulative Consumer Surplus (7) 20 1 20 20 0 0 0 14 2 28 34 6 6 6 10 3 30 44 14 8 14 6 4 24 50 26 12 26 4 5 20 54 34 8 34 3 6 18 57 39 5 39 2 7 14 59 45 6 45 13 or satisfaction which he derives from that pound is as great as that which he could obtain by spending 20s., on other things. When the price falls to l4s., he could, if he chose, continue to buy only one pound. He would then get for l4s. what was worth to him at least 20s.; and he will obtain a surplus satisfaction worth to him at least 6s., or in other words a consumers’ surplus thus showing that he regards it as worth to him at least 20s. + 14s.; i.e. 34s. His surplus satisfaction is at all events not dimin ished by buying it, but remains worth at least 6s. to him. The total utility of the two pounds is worth at least 34s., his consumers’ surplus is at least 6s. j In addition, column (5) equals column (3) minus column i (4). ' Finally, column (6) is the difference in price times the quantity purchased; it is the extra consumer surplus at that price and quantity. For example, the second entry, 6, is obtained as follows: (20 - 14) x 1 6; similarly, the third entry, 8, is gotten as follows: (14 - 10) x 2 = 8; and so on. Finally, column (7) represents the partial sums of column (6) and is shown to be equal to column (5) as a check. The significance of the consumers’ surplus is further reinforced by Marshall as follows: l6Ibid. | The significance of the condition that he buys the second pound of his own free choice is shown by the consideration that if the price of l4s. had been offered to him on the condition that he took two pounds, he would then have to elect between taking one pound for 20s. or taking two pounds for 28s.; and then his taking two pounds would not have proved that he thought the second pound worth more than 8s. to him. But as it is, he takes a second pound paying l4s. uncon ditionally for it; and that prives„that it is worth at least l4s. to him. ' The first criticism that was voiced against this measure of satisfaction is the inherent assumption that i 18 1 the marginal utility of money remains constant. The second criticism levied against Marshall is ic his assumption that the individual surpluses are additive.' Specifically, Marshall states: We may not pass from the demand of an individual to that of a market. If we neglect for the moment the fact that the same sum of money represents different amounts of pleasure to different people, we may measure the surplus satisfaction which the sale of tea affords, say, in the London market, by the aggregate of the sums by which the prices shown in a complete list of demand prices for tea 17Ibid., p. 435- 1^Rima, p. 256. See also David M. Winch, "Con sumer's Surplus and the Compensation Principle," The American Economic Review, LV (June, 1965), 395-423. 1^Winch, p. 406. See also Rima, p. 256. 20 exceeds its selling price. Still, in spite of these shortcomings, consumer 21 surplus has remained a commonly used measure of welfare. Pigou In 1912, Pigou, in his Wealth and Welfare, pos tulated that the effective segregation (and price charging) of a market into subgroups of equal elasticities depended on the non-transferability of the commodity in question 22 between the markets. In addition to identifying the conditions necessary for price discrimination, Pigou 2 ^ also investigated its welfare aspects. 3 Thus only 68 years after Dupuit1s groundwork, we find the skeleton 24 of a formal structure for price discrimination erected. 20Marshall, p. 436. 21Gerhard Tintner and Malvika Patel, "Evaluation of Indian Fertilizer Projects: An Application of Con sumer’s and Producer's Surplus," Journal of Farm Economics, XLVIII, No. 3 (August, 1966), 704; Eli W. Clemens, "Price Discrimination and the Multiple-Product Firm," Review of Economic Studies (1950-1951)» reprinted in Readings in Industrial Organization and Public Policy (Homewood: Richard D. Irwin, Inc., 1958), p. 274; Winch, pp. 395— 423. 22Rima, p. 292. 2^Rima, p. 293* p ji Relative to the 1838-1933 span for Connot’s marginal analysis. 16 The remainder of this section is a fuller expo sition of the two ideas stated above. The source is Pigou’s later work entitled The Economics of Welfare, Chapter XVII, entitled "Discriminating Monopoly. This second book is only a revision of Pigou’s 1912 i 26 work. In sections one and two of Chapter XVII, Pigou states: Up to this point we have supposed that monopolisation, when it occurs, will be of the simple form which does not involve discrimination of prices as between different customers. We have now to observe that this variety of monopolisation is not the only pos sible sort. Discriminating power will sometimes exist alongside of monopolis tic power, and, when it does, the results are affected. It is, therefore, impor tant to determine the circumstances in which, and the degree to which, monopo lists are able to exercise, and find advantage in exercising, this power. The conditions are most favourable to discrimination, that is to say, dis crimination will yield most advantage to the monopolist, when the demand price for any unit of a commodity is independent of the price of sale of every other unit.2? 2^A. C. Pigou, The Economics of Welfare (4th ed; London: Macmillian and Co., Ltd., 1950), p. 275* P f i Ben B. Seligman, Main Currents in Modern Eco- nomics (New York: The Free Press of Glencoe, 1963)> P- 477. 27Pigou, p. 275. Furthermore, the independence of price has two implications. The fact is that no unit sold in one mar ket may be resold in another. The second implication is that no unit of demand in one market may be trans ferred. 28 In the first case, a doctor's application of medicine, for example, prohibits any transfer. The second implication is more subtle; here, it is assumed that a wealthy patient does not demand the same treatment as a poor one. Pigou further notes that product differentiation is an artificial method of creating different and hence pQ non-transferable demands. The author points out how ever that the practice of strong discrimination could result in demand transfers. For example, should off season hotel rates be lowered enough, some vacationers, perhaps, would reschedule their vacations. In addition, bargaining as a remedy is ruled out because under these circumstances the monopolist faces many buyers, and the latter accept the set price. Also, Pigou makes clear and full use of the concept of a consumer surplus as a measure of the degree of 28Ibid., p. 276. 29Ibid., p. 277. 18 discrimination. Both points are illustrated in the following quotation: The loss of an individual custo mer's purchase means so much less to the monopolistic seller than to any one of the many monopolistic purchasers that, apart from combination among purchasers, all of them will almost certainly ac cept the monopolistic seller's price. They will recognise that it is useless to stand out in the hope of bluffing a concession, and will buy what is offered, so long as the terms demanded from them leave to them any consumers' surplus.3° With these preliminary notions, the author proceeds to identify the three common types of price discriminations. The first degree price discriminator simply takes all of the consumers' surplus. The second degree price discriminator charges different prices for different quantities. Finally, when a monopolist can break up the buyers into groups with similar demand and prevent trans fer of the goods, we have a third degree price discrimina tor. Again, the following quotation delineates the classi fication : A first degree would involve the charge of a different price against all the different units of commodity, in such wise that the price exacted for each was equal to the demand price for it, and no consumers' surplus was left to the buyers. A second degree would obtain if a monopolist were able to make n sepa rate prices, in such wise that all units 3°Ibid. , p. 278. 19 with a demand price greater than x were sold at a price x, all with a demand price less than x and greater than y at a price y, and so on. A third degree would obtain if the monopolist were able to distinguish among his customers n different groups, separated from one another more or less by some practicable mark, and could charge a separate monopoly price to the members of each group.31 Figure 3 illustrates first and second degree price discrimination. It. is apparent from this illus tration that some consumer surplus (the shaded areas) is left to the buyer in the case of second degree price discrimination. It should be noted that as the number of prices increases, this case approaches first degree 32 price discrimination. But Pigou points out that since the identical demand schedules required for first degree price dis crimination are unlikely to occur, this type is of aca- 33 demic interest only. Similarly, the author feels that second degree price discrimination cannot be introduced 3 "... except in extraordinary circumstances." These considerations lead him to conclude that the most 31Ibid.', p. 279. 32Ibid., p. 284. 33Ibid., p. 280. 311 Ibid. , p. 281. 20 Figure 3 PRICE DISCRIMINATION 21 | prevalent form of price discrimination is the third degree type.35 Robinson Professor Robinson defines "price discrimination" as: The act of selling the same article, produced under a single control, at different prices to different buyers„g is known as "price discrimination." Furthermore, the distinction between a unit of demand and a unit sold has been dropped by Robinson (and by most subsequent economists writing in this area). Also, the author clearly indicates that only in imperfect markets can price discrimination take effect. Specifically, the more elastic the demand, the less the price differential ■ 7 7 due to discrimination. In addition, Robinson stresses the importance of keeping the markets separate, agree ments on tariffs providing some of the means. Finally, an added departure from Pigou is Robin son's renaming of the various types of discrimination as follows. The first degree type is referred to as 35Ibid., pp. 281-282. Joan Robinson, The Economics of Imperfect Com- petition (London: Macmillan and Company, Ltd., 1964), p. 179. 37Ibid., p. 180. 22 qO "perfect discrimination." The third degree type is simply referred to as price discrimination. Lastly, though no change of name is applied to the second degree type, the latter only receives a brief coverage in a 39 footnote. Equilibrium prices and outputs In section 2 of Chapter XV, the author turns to the determination of prices and output of a two market (third) degree price discriminator. The graphical analy sis is briefly outlined below. Let and D2 be the demand schedule in markets I and II respectively. Let their corresponding marginal revenue functions by MR^ and MR2> We construct the aggregate demand AD and the aggregate marginal revenue curve AMR by adding and D2 and MR^ to MR2 at each price level. Consequently, MC, the marginal cost curve of the total output is drawn in. And at MC = AMR, we obtain the total quantity sold Q,p as well as the components Q.^ and Q2. Finally, the prices in each market are read off the demand schedules and D2 for their respective quantities and Q2 yielding qO Robinson feels that Pigou did not quite develop a valid first degree price discriminator. See Robinson, p. 187, footnote 1. ^ Ibid. , p. 187, footnote 1. 23 MR^ = MR,, = MC. Figure 4 illustrates the method. ^ A special case arises when one of the markets is perfectly competitive. Such a situation could be facing a firm operating in two countries. Nonetheless the general equilibrium conditions, MR^ = MR£ = MC, are found to be 41 equally applicable. In section 5» the author analyzes the effect on output and prices when a simple monopolist turns price discriminator. Briefly, Robinson states: Suppose that a monopolist is sel ling his commodity at a single price, and that he then discovers that discrimination between the two markets is possible, every thing else remaining the same. He must now decide in what way it will be profitable to alter the price in the two markets.42 After a rather cumbersome geometric proof, the author concludes that: It is possible to establish the fact that total output under discrimination will be greater or less than under simple monopoly according as the more elastic of the demand curves in the separate mar kets is more or less concave than the less elastic demand curve; and that the total output will be the same if the demand curves are straight lines, or indeed in any other case in which the concavities are equal. 2t0Ibid. , pp. 182-183. ^1Ibid., p. 184. ^2Ibid., p. 189. ^Ibid. , p. 190. Figure 4 EQUILIBRIUM OUTPUT AND PRICES P 1 2 •AD MR MR Q 25 | ! The variations in output can be visualized with i i i i the aid of Figure 5* In this figure, D is the demand schedule facing a simple monopolist. For example, this might represent the demand for a physician's services when prices are independent of income. The average revenue curve of the discriminating monopolist, AP, is lie the path of all points, P^ = \ (P1 + P2)» Since P1 >- P2, the curve AP must be to the right of the curve D. If the average cost curve, AC, lies between D and AP, there is no output for a simple monopolist, whereas there is some output for a discrimina te ting one. Thus, Robinson concludes: The above analysis suggests that on the whole it is more likely that the introduction of price discrimination will increase output than that it will reduce it .47 ^Donald S. Watson, Price Theory and Its Use (Boston: Houghton Mifflin Company, 1963), PP* 320-321. ^ Ibid. , p. 320. This example differs from Robinson's (Robinson, p. 197) in two ways. First, AP and D join at some price, P, greater than P-i. Secondly, Robinson's demand curves are not linear. Simplifications were carried out for exposition purposes only. 117 'Robinson, p. 201. 26 | *• *' ! Figure 5 OUTPUT WITH PRICE DISCRIMINATION P 1 AP 2 AC Q 27 These findings are in agreement with Pigou’s: Hence, it follows that, under decreasing supply price, monopoly plus discrimination of the third degree may raise output above the competitive amount, and is more likely to do this the more numerous are the mar kets between which discrimination can be made.^8 It must be pointed out, however, that though the conclu sions are similar, the premises are different. Speci fically, Robinson writes: Professor Pigou states that in con ditions in which there would be some sales in each market under a single price "there is no adequate ground for expecting either that output under discriminating monopoly .... will exceed, or that it will fall short of, output under simple monopoly" (Pigou, p. 286). But he was led to this conclusion because his precise analysis deals only with straight-line demand curves and does not enable him to isolate the con ditions in which discrimination will in crease or reduce output. In a passage which appears inconsis tent with this one, he reasons that be cause perfect discrimination must increase output ordinary discrimination is likely to increase output (Pigou, p. 287), and he argues that it will be more likely to do so the larger are the number of separate markets in which the monopolist can sell. But, as we have seen, the result depends not upon the number of markets but upon the relative concavities of the separate demand curves.^9 48Pigou, p. 287. ^Robinson, p. 201. The author of this dissertation feels that Robinson's reasons are the correct ones. Leontief The problem that Leontief addresses himself to is best stated by his introductory remark: The established theory of discrimina tion deals only with the case of Unlimited Discrimination. The seller — a monopolist — is assumed to be in a position to sub divide the potential buyers of his product into two or more independent markets and is supposed to be free to establish in each separate market any price he may choose. No outside limitation whatsoever is imposed upon the comparative height of the prices quoted simultaneously to the different groups of purchasers.51 The theory however, is not in line with common practices; specifically, the author states: An examination of various actual in stances of price discrimination shows, however, that in contrast to the common theoretical assumptions mentioned above, a great many, not to say most, discrimina ting monopolists are able to exercise their power of price differentiation (-2 only within certain rigidly defined limits. Empirically, the author supports his contentions 5°W. W. Leontief, "The Theory of Limited and Unlimited Discrimination," Quarterly Journal of Economics, LIV (1939-19*10), *190-501. 51Ibid., p. 490. 52Ibid. with several cases. First cited is the case of "dumping," I where the difference between the domestic and foreign prices is bound by the protective import duty. Similarly, inter-regional price discrimination is limited by the transportation costs. In still other cases, the price differential might simply be limited by administrative regulations. All these cases belong to what Leontief calls • unlimited price discrimination. In the remainder of his < I I paper, the author investigates the effects on profits j and prices when a two market price discriminator is ! subject to the constraint P - ] _ “ ^2 = a> w^ere anc^ ^2 are the prices and a is the limiting range. Since the mathematical aspect of the Leontief analysis has been treated elsewhere, this section of the review restricts 54 itself to an outline of the findings. Given the more rigid constraint, - P2 = a, the monopolist endeavors to maximize profits. Applying differentiating techniques, the author first finds that for profit maximization the sum of the "marginal profi tabilities"'^ in each market add up to zero. Secondly, 5 3 This figure should be adjusted by the shipping cost, however. 54 See Chapter III of this dissertation. "^Leontief, p. 493- 30 ; i I the rate of change of profits with respect to the con- ! straint is found to vary positively with the marginal profitability of the higher priced market, hence negatively with the marginal profitability of the lower priced 56 market. The reconciliation of Leontief*s terminology 57 with the one of previous models is presented as follows. Discrimination, or discrimination of the third degree, is equivalent to Leontief*s unlimited discrimina tion, e.g. no constraint. The term "no discrimination,” I i refers to the case where ~ ^2 ~ ^' Leontief*s limited discrimination, e.g. P^ - = a, is novel, and hence has no counterpart in previous models. With the nomenclature problem resolved, the author directs his attention to the following theorem: The well-known proposition which emerges from such a comparison states that the introduction of (unlimited) discrimination in place of a non-discrimin- atory policy might reduce the prices in both markets, provided the particular commodity is produced at decreasing mar ginal costs.5o The word, "provided," in the last sentence implies 56 The equilibrium conditions found in this dis sertation are reconciled with those of Professor Leontief (see Chapter III). •^Leontief, p. ^96. 58ibid. 31 that decreasing marginal costs are a necessary condition for possible price reductions. Professor Leontief denies this and proceeds to prove the following two propositions: A. Decreasing marginal costs of produc tion constitute a necessary condition for increase of both market prices in transition from a single-price policy to a policy of unlimited discrimination, pro vided the shapes of the demand curves and of the cost curve involved are such that there exists only one single, optimum amount of discrimination which could maxi mize (in the absolute and the relative sense) the profits of the monopolist (the "profit hill" has only one "summit"). B. If the shapes of these curves are such that the monopoly profit can be maximized (in the absolute or the rela tive sense) at two or more different price combinations, i.e., if the "profit hill" has more than one "summit," de creasing marginal costs do not consti tute a necessary condition of the des- cribed price increase.59 Following the proofs, the author replaces the previously accepted proposition by the following less restrictive theorem: Introduction of "limited discrima- tion" in place of "no discrimination" can lead to a price reduction in both markets even if the monopolist produces at increasing marginal costs.60 Reference to this theorem is made in the conclusion 59Ibid., p. 497. fin Ibid, , p. 501. 32 of this dissertation. Clemens The determination of prices in a multiple-product firm was carried out ingeniously through the techniques of price discrimination by Professor Eli Clemens. In his well known article, the author introduces the problem thusly: The problem of the multiple-product firm has lain in virtual neglect on the thres hold of the theory of monopolistic (or im perfect) competition since the pioneering efforts of Chamberlin and Joan Robinson some seventeen or eighteen years ago. Close ly related to this problem is the problem of price discrimination or the price line. The significance of both is apparent, since it is probably impossible to find in the whole of our economy a single firm that sells a single product at a single price. This is theoretically explainable by the bact that the conventional single-product firm that is presumably in equilibrium when mar ginal revenue is equal to marginal cost is not in equilibrium if it can serve the re maining protion of the demand curve at a price greater than marginal cost without adversely disturbing its existing market, or, more commonly, if there is any accessible * 1 Eli W. Clemens, "Price Discrimination and the Multiple-Product Firm," The Review of Economic Studies, XIX (1950-1951)s 1-11. ''Reprinted with Alterations," in Readings in Industrial Organization and Public Policy (Homewood7 Illinois: Richard D. Irwin, Inc., 195&), pp. 262-276. 1 I market for which it can produce with its unused capacity at a price above marginal cost. The first situation, price dis crimination, differs only slightly from the second, multiple-product production; to gether they constitute the terrain of the firm's activities.62 The fundamental assumption underlying the analysis is that the firm sells the "capacity to produce" rather than just products. 2 The decision to produce a new product, hence a reduction in excess capacity, will be made as long as the potential price is greater than the marginal cost. If we extent this process to several successive goods, their production is carried out "... to the point where marginal cost approximately equals price 64 in the least profitable market." Finally, the author theorizes that the removal of the artificial intra-firm product differentiation techniques renders the distinction between price discrim- ination and multiple-product production irrelevant. Hence, the author states: The problem of the multiple-product 62Ibid., pp. 262-263. 63Ibid., p. 263. 6**Ibid. , p. 264. 65Ibid., p. 264-265. 34 firm can then be treated simply as a problem in price discrimination, and Joan Robinson’s well-known analysis of what Pigou calls discrimination of the third degree will lend itself with certain reservations to the consideration of the problem.66 Clemens raises several objections to the Robinsonian analysis, however.The first objection is that produc tion is not extended to where marginal cost is equal to demand. The second objection is that Robinson does not envisage quantity discounts. Thirdly, the identification of the change in marginal cost associated with a specific change, rather than a general change in output, cannot be made through Robinson's technique. In light of these difficulties, Professor Clemens presents his alternative analysis which is applicable to both price discrimination and multiple-product pricing. A brief description of f \ R the author's procedure follows. The firm is assumed at first to produce one product under conditions of 60 to 70 percent of capacity. Still for that product, marginal cost equals marginal revenue. A second product can be produced with the idle capacity if its demand is above marginal cost. The 66Ibid., p. 265. 67Ibid., p. 266. ^ Ibid. , pp. 266-267. ! 35 same principle is applied to subsequent products until the last product where demand is approximately equal to marginal cost. The situation is illustrated in Figure 6 which is reproduced from the article being reviewed. Profits are maximized when production is dis tributed among the five markets in such a way that all marginal revenues are equal (equal marginal line EMR) and they in turn are equal to marginal cost of the whole 6 9 output. As can be seen, the lowest price is just above marginal cost. Also, in light of the assumption of suc cessive entry, each new market is more elastic than the preceding one. This in turn implies that prices and profits go down with each additional product expansion. There is one further ramification, which is best put by the author. If this last market is one of perfect elasticity, equivalence is per fect. It also follows that if any single market demand served by the firm is one of infinite elasticity, it becomes the marginal market. Any marginal market of less than infinite elasticity leaves open the possibility, although not the absolute necessity, of some unserved market in which price is greater than marginal cost. The procedure presented by Clemens makes clear a dynamic 69Ibid., p. 267. 7°Ibid., p. 268. Price-Cost 36 Figure 6 MULTIPLE PRODUCT PRICING AND OUTPUT MC EMR 0 0 0 0 0 0 OUTPUT SOURCE: Clemens, p. 267. 37 expansion process into lesser and lesser profitable mar- 71 kets. The author indicates that this is supported by 72 empirical evidence in several instances. First, he notes that practically all firms produce some products at almost no profit. In some cases, this is done to keep the organization intact. Specifically, this has been observed in railway and utilities industries. There is one last point where this analysis differs from Robinson’s presentation. This is the application by the firm of sliding scale of quantity discounts. Thus, the author states: In such an instance the monopolist might select a series of prices along the demand curve, each price being applicable to all customers willing to pay it and unwilling to pay the next higher price. In short, the demand curve would be "segmentized." The more prices that could be established, the greater would be the monopolist’s profits. In the extreme and limiting case the monopo list would obtain the full demand price for each unit of output by establishing an almost infinite series of prices. Con sumers' surplus would be entirely eliminated and the demand curve would become the marginal revenue curve.73 Such cases are well reflected by the pricing 71Ibid., p. 271. 72Ibid. 73Ibid., p. 274. practice of gas and electric utilities where price dis crimination is not only applied to classes of consumers, 7i| but to quantity purchased as well. The results of Clemens’ analysis are summarized in Figure 7- First, the realized average revenue curve lies above the demand curve. Thus, profits are realizable even if the average cost curve lies above the demand curve. Second, this divergence increases with the increase segmentation of the demand curve. The pursuit 75 of maximum profits are hence achieved in two ways: (1) by the invasion of new markets; and (2) by splintering the existing markets. The foregoing analysis is best summarized by the author as follows: The distinction between a producer selling a single product at different prices and one selling different products in varying markets at differing percen tages of profit is a distinction of degree only.76 And lastly: The theory of price discrimination must be viewed as the heart of price- cost theory rather than a peripheral 74 ' Block-rate scheduling, for example, Ibid. 75Ibid., p. 275. 76Ibid., p. 276. Figure 7 AVERAGE REVENUE AND REALIZED AVERAGE REVENUE Price AR 0 0 0 0 OUTPUT SOURCE: Clemens, p. 275* case. The firm that does not discriminate in its pricing policy, or differentiate in its product line, or invade new mar kets, dies in the competitive struggle — and "business management does not commit suicide."77 77Ibid. CHAPTER III THEORETICAL STRUCTURE The first section of this Chapter examines the general equilibrium characteristics of a constrained price discriminator facing two markets. By general, we mean with unspecified demand and cost functions. Also, for each market, we analyze the determination of price and output. The second section presents the equilibrium characteris tics for specific demand and cost functions.- Similarly, the determination of prices and quantities are also exam ined. In addition, a simple cost allocation algorithm is presented. Finally, this chapter closes with a brief summary. Theoretical Models This section consists of three parts. Part one presents a theoretical model together with the underlying assumptions. In the case of the discount demand curve a rationale is offered for the inclusion of the discount 41 42 rate in the function. Part two discusses equilibrium from a static and comparative static standpoint. Part three covers the differences in equilibrium between a constrained and an unconstrained firm. A brief summary follows part three. The Models The models consist of the following relationships Qd = Qd(P,,d) with ^ d < 0 and (1) 6Pd 6Qd •jr- varies with d od 6Q„ Qr - Qr(Pr) with < 0 (2) Pd = (1 - d) P^ with 0 _< d < 1 (3) Q = Qr + Qd W C = C(Q) (5) where Qd is the discount demand Pd is the discount price d is the discount rate Qr is the regular demand P^ is the regular price Q is the total quantity demanded 43 C is the average cost function For the purpose of this dissertation, the following situation is stipulated. A monopolist faces one demand curve Q^. For reasons beyond our scope, he discounts the price, by d(Pr) and presents this new price, - d(Pr) = (1 - d)Pr to a small group. The latter takes the price, P^ and purchase quantity Q^. We assume that this small group is new to the product. That is to say, the new group was not part of the demand Q^. Figure 8 illustrates the case. Figure 8 MARKET CREATION P MC d MR 44 At first, is very small relative to Qr* In fact, does not enter the profit maximization decision. Since d induced Qd, d enters the demand function.1 With time, Qd may grow; when reaches relatively large values and if < cost, then the firm must take into account three factors or go out of business. These factors are: (1) Total conceived revenue changes from prQr to TR = Pr Q r + PdQ a (2) Total real cost goes from C(Qr) to TC = C(Qr + Qd) (3) Although initially a voluntary act, the dis count has now become a constant, namely Pa - C i - d ) P p Now, the monopolist maximizes profits subject to the constraint = (1 - d) P^. He now makes more 2 profits than in the first instance. True, he could even 1Tibor Scitovsky, "Some Consequence of Judging Quality by Price," Review of Economic Studies, XII, No. 2 (1945), p. 101. This follows from the fact that he now includes in the optimization process the discount output Qr in addition to Q^. 45 make more profits if he set MR = MC in each market; but this case is not the issue in this dissertation. Finally, it should be noted that several discount rate groups may coexist. Thus, Q. and Q, may have discount rates al 2 d1 and d2 respectively. For exposition purposes, only the single case will be analyzed. Equilibrium Considerations Given the discount rate d, the prices are not | independent. In choosing P , the monopolist simultan eously chooses ' Profits (it) equals f t = 7r(PrSPd) ( 6) For maximum, we have d-rr _ 6ft + 6tt ^ d 0 (7) But from equation (3) (8) Hence, equation (7) becomes dP ft^ + (1 - d)TT2 = 0 (9) r where f t _ 6jt 1 " 6 P. and ftp = 6tt r 46 For the second order condition for a maximum, we differentiate equation (9) with respect to P as follows: ,2 dP, dP, " - It,, + + "on + IT,, 35- <0 (10) dPr where 2 " "11 "12dP "21 "22 dPr _ *2 , = — V » "12 = § jr. 11 6Pr {PrSPd 9 x2 *2^ _ O IT ^ 0_TT__ *21 = <SPd<SPr 3 ^22 6p 2 Making use of equation (8) and noting that 12 = 21s • 3 equation (10) reduces to: *11 + + 7T22^”C -^ ® (11) ^As noted in the introduction, the general con straint is P^ = (1 - d)Pr - s, where d and s are given. Hence, (dPd/dPr) = (1-d). Therefore, equations (9) and (11) are still valid for this more general case, Furthermore, if we let d = 0, equations (9) and (11) respectively reduce to: u + ir2 - 0 *1 + 2*12 + 7T22 < 0 which are identical to Leontief’s system. 47 Finally, in order to observe the behavior of and with respect to d , we differentiate equations (9) and (3) as follows: dP dP, dP dPd *11 dd” + *12“dd + *21 “dd + *22 “dd = 0 (12^ and dP. dP^, /nov — — = (1 - d) — — - p (13) dd a' dd r Combining terms in equation (12) and rewriting equation (13) yields equation (14) and (15) respectively. dPr dPd (*11 + *21 ^ + (*12 + *22^ ~dd = 0 dP dP. (1- d) = Pr (15> We have two equations in two unknowns, and the solutions are: dFr _ Pr(*12 * *22^ d d ( 1 - d ) (tt12 + tt2 2 ) - (tt11 + tt2 1 ) d P d = p r ( * n + TTgl^ ______________ f T 7 v dd ( l - d ) ( ir12 + tt2 2 ) - (tti;l + ir2 1 ) K I J In a similar fashion, profits tt in equation (6) vary according to: — = TT d P r + TT d ? d dd *1 dd ^2 dd (l8) Substituting equation (15) into equation (18) yields: , dP dP dd = *1 “dd + ^2 ^ 1_d^ “df “ Pr} Prom equation (9) we have: ir2(l-d) = -tt1 Hence, equation (19) reduces to: = -P i t = > r 1 k (20) dd r 2 (1 - d) Thus, if > 0, then an increase in d raises total pro fits. Or if ir^ <0, profits decrease. Constrained Versus Unconstrained Equilibrium For profit maximization under the constraint P^ = (1-d) Pr, we have from page 44 n ■ Pr • Qr + Pa • «a - C(Qr + Qd) (21) 49 l i For the lagrangian method, we have: V(Qr, Qd, X) = PrQr + PdQd - C(Q) - X(Pd - (l-d)Pr ) (22) For a constrained maximum, first order conditions are: If = Pr + QrS^ “ fo“ + (1"d)<SPr = 0 (23) 6Qr r r6Qr 6Q 6Qr § 1 = P + Q f!d . «C «Q _ !!d = o (24) 6Q d Qd Q Qd Qd d II = Pd + (l-d)Pr =0 C25) Second order conditions have been discussed above. By definition: 6Q 6P ■ r . r «Pr «Qr "r and 6Qd . 6Pd -e_ (26) 6pd 6Qd -d -e* (27) 4 In the Marshallian demand curve, P is the de pendent variable. Since our graphs will be shown on this basis, we formed V(Qr, Qd, X) and not V(P , Pd»X ). The results would of course be identical. 50 Substituting equations (26) and (27) into equations (23) and (24) yields: Pp - !e - HC - X(l-d) Jk . 0 (28) P P ■ r - MC + ^ r r = 0 ^ d ed % ed Where MC = ^ since = 4^— = 1, or MC is the dQ 6 Qr 6Qd 9 marginal cost of total output. Rewriting equations (28) and (29) yields: P MR - MC = A(l-d) ^ (30) ^r r MRd - MC - - _i (3!) d d where MRi ■ pi(1 - ¥7) Dividing equation (30) by (31) and noting that ((l-d)Pr/Pd) = 1, we have: MRr - MC = MRd - MC Qr er (32) if MRd t MC . (33) 51 For the constrained monopolist, the inequality holds for the following reason: only if X = 0, (an unconstrained firm), will marginal revenue equal marginal cost in equations (30) and (31). Specifically, if X > 0 (34) and since Pr > 0 by definition (35) Qr > 0 by definition (36) er > 0 for a normal good (37) then equation (30) yields MR^ > MC. Conversely, if X < 0, equation (30) yields MR^. < MC. Hence, MR^ f MC for the constrained monopolist. We note further that the departure from MR = MC (38) is not only a function of the ratios of the market shares (Q./Q) r (39) but also of the ratios of the elasticities. Algebraic Models In this section we first specify the theoretical models of the previous section. Secondly, we solve the first order conditions in equations (25), (28) and (29). 52 Thirdly, a method to allocate costs is developed, in order to compute profit per market. Finally, we isolate the regular market and compute the unconstrained equili brium values in order to analyze the position of the con sumer . The Models Regular demand: Qr = bo + blPr (1|0) Hence, the inverse demand is: (HI) bl bl 5 Discount demand: Qd = aQ + a1Pd + a2(l-d) + a^Cl-d)2 (42) The first order term follows from our assump tion. The second order term follows from the following observation by Professor Scitovsky: The commodity offered at a lower price than competing commodities will be both more attractive to the consumer on account of its greater cheapiness and less attractive on g account of its suspected inferior quality. ^Scitovsky, p. 101. 6Ibid. This induces Scitovsky to call price a "double-edged" weapon. This "double-edged price" reduction may be repre sented mathematically by a parabola. Figure 9 illustrates the point: Figure 9 DOUBLE-EDGED WEAPON range B range A In range A, as d increases, so does the quantity deman ded; a bargain is at hand. In range B, demand decreases; a poor quality good is offered as is reflected by the price which is low. It should be noted that the empirical range of d, over which the fit was made did not include the origin nor the point d = 1. Hence, 7 this figure reflects the more restrictive range. Thus (43) letting AO = -a a — (1-d)2, equation (43) 1 becomes: P d (44) Average cost function; C = cQ + c^Q + c2Q2, with cQ > 0, c2 > 0, and c^ < 0 where Q = Q + Qd (45) Thus, total costs, TC, are TC = cqQ + c-jQ2 + c2Q3 (46) 7 'See footnote 5, Chapter V. Hence, marginal costs, MC, are MC = cQ + 2c1Q + 3c2Q2 Equilibrium Prices and Output Prom equation (42), we have: e = -a d 1 Qd Hence V d = -aipd Similarly, from equation (40) h Pr er - -bi Therefore Q e = -b,P r r 1 r Dividing equations (49) by (51) yields V d . V d ®rer blPr but = d “d)» hence 56 Substituting this result into equation (32) yields: MR - MC a. MR. - MC = b7 (51 *) d 1 expanding equation (54) we obtain: a.n - (MR^ - MC) = (MRd - MC) %=- (1-d) (55) where from equation (4l) MRr = ' ^ + k (56) Prom equation (47) MC = c + 2c. Q + 3c0Q2 (57) O 1 2 and from equation (44) MR. = AO + (58) d a^ d Expanding equation (54) S i 3. -MR + MC = r-—■ (l-d)MR. - r~~ (l-d)MC (59) r b.^ d b^ Collecting the terms yields: -MR - (l-d)MR. = - (l-d)MC - MC (60) r b^ d I 57 Hence, cl £ L MRr + F - (1~d) MRd = (F" (1“d) + 1) MC (6l) Furthermore, the constraint is rewritten as follows: Pd = rPr (62) where r = 1-d Substituting equations (41) and (44) into equation (62) yields: A0 + ^ Qd = r S f + V (63) expanding and collecting terms yields: gE Q r - _ i . Q d . A O + (6#) Also Qr + Qd = Q- (g5) We have two equations and three unknowns. Letting Q be a parameter, Qr and Qd are solved as follows: = A(J + rbo/b1 + Q/ai (6fi) r/bx + l/a1 58 Qd = -AO + rbo + (r/b^Q (6?) r/b1 + l/a1 For manipulation purposes, equations (66) and (67) are rewritten as follows: Qr = fl + f2Q (68) Qd = f3 + fjjQ (69) From equation (68), equation (56) becomes: MRr ‘ + bf (fl + f2e> (70) Similarly, from equation (69), equation (58) becomes: MRd = AO + ^ (f3 + f^Q) (71) Substituting equations (70), (71) and (57) into equation (55), we obtain _ 2 al ^ ^*2 2 + bf fi + rA0 + 2EYr+ 2 b f + bf cl 3, 3, (^ r + l)c0 + 2C;l( ^r+l)Q +3c2 (^r+l)Q2 (72) This equation is a quadratic in Q: Do + D1Q + D2Q2 = 0 (73) 59 Where D o o (7*0 (75) D 2 "3g2 ^ r + (76) Hence, for a given discount rate, d, where (d+r = 1), zu where Qgi 0 Having solved for Qg, we compute Qre and Qrd from equations (68) and (69). Pre and Pdg are com puted from, equations (4l) and (43). Finally, total costs are computed in equation (46). (77) 60 Allocation of Costs In order to compute the profits per market, we need to know the cost per market. Unfortunately, a U- shaped cost function is not subject to linear transfor mations, such as T(Qr + Qd) = T(Qr) + T(). To put it differently, a non-linear cost function states that the cost in one market is a function of the cost in the i other market. To circumvent this difficulty, we will stipulate that after equilibrium values of and are obtained, total cost will be allocated according to the shares of the markets thusly: (78) e (79) where TCre = total cost in the regular market, and TC, = total cost in the discount market. Finally, profits are computed as follows: re (80) (81) 61 Constrained Versus Unconstrained Prices In the first section of this chapter we indicated how the discount price, P^, might have induced the crea tion of the discount market. Under this assumption, in order to remove the constraint, we must remove the discount market altogether. That is to say, the latter would not exist without d (this is not analogous to setting d = 0 because at this value, some demand, Q^, might still exist algebraically, yet logically, there should be none). With a U-shaped average cost curve, the elimination of Qr makes a difference. Thus, on the downward part of the curve, setting = 0 in C(Qr+Qd) results in higher prices. This follows from the fact that the coefficient of Q is negative and is pulling costs down. On the upward portion of the curve, the p Q term is pulling up with a positive coefficient, hence the converse is true. This is shown below. For unconstrained profit maximization, we have MRr = MCr (82) The subscript in MC^ indicates that the marginal cost is computed for Qr only. Substituting equations (56) and (57) into equation (82) yields: 62 - E7 + ^ ®r " °o + 2olQr + 3e2«? (83) Combining these terms yields °o + + C2ol - b^)Qr + 3o2Qr (81|) Again, we have a second order equation, and solving for Q yields: -(2c1/2/b1) +N y{2c1-2/b1)2 -12(c0+bo/b1)c2 Q = ----------------------------------------------------- 2c2 We define Q_ as the unconstrained output. 3 We note that since Cq > 0, bQ > 0, c2 > 0, and b^ < 0, the second term under the radical is positive. Furthermore, since c^ < 0, the expression, -(2^-2/b^) is positive. Hence, >_ 0. With the unconstrained output, Q , we obtain the unconstrained price, P , 3 3 and the ratio (P0-PM/P0). In chapter IV we present an 8 xx 3 empirical case. It should also be noted that other obvious choices are the regulated price .and the competitive price. The former is where AC = P, eliminating excess profit; the second is where MC = P, or the competitive price. Figure 10 illustrates some of the possibilities. 63 Figure 10 VARIOUS NORMATIVE PRICES P monopoly price regulated price ,MC AC ompetitive price MR Q 64 The reason for choosing the monopoly price is that the other two are different hypotheses and not in line with Q the objective of this dissertation. Summary As a result of the preceding analysis, we see that under our assumptions, the ratio, (PQ-P_,)/PQ = d’, cl A cl comes in the long run from the discount rate d. Func tionally, we have: d' = f(d). To render this rela tionship causal, we subscript the discount rates as 9 follows: d't = f d ^ ) Hence, in the short run, d might be used as a policy tool for controlling monopoly profits or prices. This will be shown in Chapters IV and V. O For example see Harvey Averch and Leland L. Johnson, "Behavior of the Firm Under Regulatory Con straints," The American Economic Review, LII, No. 5 (December, 1962), 1052. ^The validity of such "casual dynamization" has been questioned by Gerhard Tintner, "Some Thoughts About the State of Econometrics," in The Structure of Economic Science, ed. Sherman Roy Krupp (Englewood Cliffs, New Jersey: Prentice Hall, Inc., 1966), p. 121. CHAPTER IV EMPIRICAL WORK In this Chapter, we indicate how the theoretical models of Chapter III were fitted. Specifically, we discuss the compromises that were necessary to obtain good fits. We also present difficulties that were en countered in the data proper and how these problems were resolved with the corresponding explanation for the particular method used. This Chapter is organized in three sections. The first presents the difficulties encountered in ob taining empirical discount data. We conclude this sec tion with a rationale for selecting the U. S. Post Office for our analysis. The second part discusses three types of data problems. In the last section, we present 54 models and their statistical properties distributed as follows: 24 models for Class I, 12 average cost curve models, and 18 discount demand models. We close this section with a brief evaluation and summary of the 65 66 empirical models. The Empirical Firm Letters were sent out to ten large firms to obtain cost, revenue and discount data. The same message was carried by all the replies: no data. It is possible that the data were present but that the firms were reluctant to divulge them for fear of repercussions in light of the Nader revolution. In a similar fashion, Moody’s Industrial Manual, Fortune Magazine, and Standard and Poor’s were equally devoid of discount data. The United States Post Office, on the other hand, though not a profit maximizing firm, has been under attack for years for the discount it has been granting to all classes except first class.'*' Table 2 shows selective years of total revenue and total cost for the major four classes of mail. It is apparent that only first class mail pays for itself. A study by Baratz indicates that 2 even marginal costs are not covered in other classes. ■*"J. George Butler, "Toward Postal Reorganization," Christian Century, LXXXVII, No. 4 (January 28, 1970), 104-108 and David Sanford, "Post Office Blues," New Republic, CLXII, No. 12 (March 21, 1970), 19-22. ^Morton S. Baratz, "Cost Behavior and Pricing Policy in the Post Office," Land Economics, XXXVIII (November, 1962), 305-314. Table 2 TOTAL REVENUE AND TOTAL COST OF MAIL BY CLASS (in millions of dollars) YEAR CLASS I Total Total revenue cost CLASS Total revenue II Total cost CLASS Total revenue III Total cost CLASS Total revenue IV Total cost 1933 319.2 249.1 19.84 108.8 50.93 79.23 100.3 132.3 1940 391.8 267.5 24.95 110.4 75.12 101.4 133.8 155.6 1950 741.2 665.2 45.00 242.3 153.7 291.6 403.8 506.0 I960 1510.1 1395.1 81.19 412.1 441.4 711.0 607.2 735-7 1969 3135-4 2898.1 147.4 620.0 781.5 1246.0 831.2 1024.0 • SOURCE: Cost ascertain reports, United States Post Office cr\ 68 Being a public agency, the data are abundant and readily obtainable. Because of these losses, various reorganization plans have been presented for the giant complex (500,000 employees). One such plan has been simply to let the U.S. Post Office become a private firm and operate under profit maximizing motives. This dissertation investigates this conjecture with one stipulation. The Post Office will be treated as a profit maximizer, but the discounting practice to second, third, and fourth class mail would be continued. Hence, from our economic standpoint, we would have a price discriminator with the constraint that: P0 = (1-cOPj where the subscript, I, means first class mail and o , means other classes; d stands for the discount rate. The logic for the hypothetical continuation of the discount practice is as follows. The same forces which would enable the concessioning out of the U.S. Post Office to private business would be able to secure, as they have in the past, discount clauses on their be half. The subsidy to farmers to this data attests to the persuasive strength of similar forces. 69 The Data This section is divided into three parts. The first covers the sources of the data, problems of accuracy, apparent inconsistencies, missing points, and the respective remedial measures. The second part covers defects in precision prevalent in published economic time series. The final part presents the accounting method used to consolidate classes II, III, and IV into the discount market. Data Problems Price, cost and volume data was gathered for the years 1933 through 1969, from the "United States Post Office Department Cost Ascertaining Reports." Over this period, two reporting systems were used. The first sys tem spans the years 1933 through 19^5* The second covers the 19^5 to the present period. Thus for 19^5» two sets of data were published for all classes of mail. Table 3 presents the ratios of the data obtained by the older ^In this context, "accuracy" means devoid of "bias" or "systematic error." Oskar Morgenstern, on the other hand, investigated the statistical or random aspects of , economic time series. Table 3 191}5 METHOD RATIOS CLASSES Method.^ revenues Method^ costs Method2 revenues Method2 costs I 1.0544 1.0003 II 1.0000 1.0010 III 1.0001 1.0005 IV 1.0005 1.0002 SOURCE: United States Post Office Cost Ascer taining Reports. method against the present one for the overlapping year 1945. For example, the present system reports Class I revenues of $615.31 million; whereas, the older method yields $648.77 million. The relative error is over five percent. Under the assumptions (a) that the newer system is more accurate and (b) that the systems have each been consistent over the years, we divided Class I revenue figures by the ratio (615.31/648.77) = 1.0544. Since all other errors are less than five percent, no further adjustments of this nature were made. Secondly, for 1943, no figures are published. Since this was the only case of missing data, we set the 1943 figures to the arithmetic average of 1942 and 1944 data points. The justification is that in light of the lack of knowledge, this simple corrective measure is probably as valid as the more sophisticated, weighted type schemes. The third correction covers cost data for 1969- The figures listed per class for that year differed from the figures of the previous year by over 50 percent. On the other hand, the total cost of running the Post Office for 1969 went from $6,681 billion in 1968 to $7,279 billion in 1969. This figure seemed logically acceptable. Hence, to replace the defective data, the 72 1969 total cost figure was allocated per class on the basis of the 1968 ratios. Again, this simple scheme was adopted in light of the limited information at hand. Significant Figures The empirical work (see the following section on empirical models) was carried out in four significant figures whenever possible. The task was greatly compli cated by recording practices of government agencies. Both the United States Statistical Abstract and the Eco nomic Indicators present the data to the same number of decimal places. Table H gives an example. These figures are chosen because they illustrate the jump from three to four significant figures. For addition purposes, this i i practice is valid. For multiplication, it is not. The reason is that the relative error in a product (X)(Y) is equal to the sum of the relative errors in each factor. Furthermore, since the relative error in a number is less than 5 x 10-n, where n is the number of signifi cant figures, the product is cut down to the least pre cise of the two numbers, and since much econometric work involves variance-covariance matrices which are made up A James B. Scarborough, Numerical Mathematical Analysis (Baltimore: The Johns Hopkins Press, 1966), p. 10. - 73 Table i | GROSS NATIONAL PRODUCT OR EXPENDITURE 1929 ~ 1966 (Billions of dollars) YEAR GNP 1939 90.5 19^0 99.7 19*11 12*1.5 19*12 157.9 SOURCE: Selective entry from: 1967 Supplement to Economic Indicators, U. S. Government Printing Office, Washington, D.C., 1967. first of products (X)(Y) and then of sums I XY, such recording practices are defective from a precision stand point . The Post Office data, on the other hand, carried at least four significant figures. Still, computations were performed with data of mixed precision in order to extract the maximum information from the data. This is further concurred by Oskar Morgenstern: Three or four digits is probably the maximum accuracy of primary data that ever needs to be considered in the vast majority of economic arguments.° Consolidation of Classes II, III, and IV into the Discount Market Since highly precise (though how accurate is unknown) cost figures were available, the "equivalent" 7 cost accounting conversion method was adopted. The following example illustrates the concept. In 1933, Class III mail cost 2.111 cents per piece, whereas Class I mail cost 2.290 cents per piece. In that year, 3459 5Ibid., p. 24. Oskar Morgenstern, On the Accuracy of Economic Observations (2nd ed.; Princeton, New Jersey: Princeton University Press, 1963), p. 45. "^Norton M. Bedford, Introduction to Modern Accounting (New York: The Ronald Press, 196b), p. 643* 75 | actual pieces of Class III mall were handled. In equi- ! valent Class I mall units, this value becomes: 2.111 3189 . u = ----- x 3^39 2.290 where u stands for pieces of equivalent Class I mall. A similar procedure was followed for Classes II and IV, using in each case the corresponding yearly cost ratio. The next step was to add yearly equivalent quan tities of the classes into the discount market quantity. Consequently, the prices for these volumes were computed from actual revenue data for each year from 1933 through 1969 in the following way: Pd = (TR2 + TR3 + TR4)/Qd where TR is the Post Office published revenue data and Pd is the discount price Finally, two time series, d and r , were developed from the price data as follows: r = 1 - d where d is the discount rate and Pr stands for the 76 regular market prices or Class I mail, and r is the fraction of payment. The following example illustrates the concept. In 1933, the discount market money price was 1.226 per equivalent piece. The same year, the Class I mail O (the regular market) price was 2.935 cents, hence, the discount rate, d , for 1933 was: 1.226 d = 1 - ----- = .4178 or 41.78 percent 2.935 Fteal prices were obtained through the Implicit GNP Price Deflator and the Personal Disposable Income Price Deflator. Both indices were shifted to 1969 = 100 in order to transform the real data to present terms. For example, the three cent postage stamp of 1933 is equal to 9.138 cents in real terms. This value should be com pared to our present six cent stamp. Empirical Models This section presents, in four parts, the results of 54 models fitted and tested by the method of least squares.^ The models are divided into three parts. --------- Actual postage was 3-0 cents, the 2.935 was com puted from the more accurate total revenue data, and used instead. ^The actual computations were performed on the 77 i The first part presents models of Class I mail or the j regular market. The second part contains models of average cost functions. The third part presents the dis count market models. In addition, each part introduces the models with a nomenclature. Also, within the part, the models have been organized into "groups” for exposi tion purposes. At the heading of each group, there appears the following notation:10 F.01 (2j 3*° = 5,29 P.05 (2j = 3,28 These are the maximum theoretical F values for the accep tance of the null hypothesis at the one percent and five percent levels of significance respectively. The numbers in parenthesis are the degrees of freedom. For example, the first model has an empirical F equal to 1390.0 ? (R = .9879) j thus it is improbable that chance alone was at work and R is accepted. Secondly, below the coeffi cients of the regressions, there appear numbers in paren thesis; these are the standard errors of the coefficients. (The Durbin Watson statistics are presented for the selected models only.) Finally, we close this section UCLA 360/91 via the BMD software package. 10This example is taken from page 81. with a brief evaluation and summary. 78 Class I Demand The models of Class I have been organized into six groups. Group 1 represents demand in prices and GNP both in money and real terms. Group 2 is on a per capita basis. Group 3 repeats Group 1 using Disposable Income (DI). Group 4 repeats Group 2 on a per capita basis. Group 5 represents demand in money and real prices only. Lastly, Group 6 repeats Group 5» again on a per capita basis. For the models of Class I Demand (Group 1 through Group 6), the following nomenclature applies: 79 Table 5 CLASS I DEMAND NOMENCLATURE SYMBOL MEANING SCALE/UNITS PSM Money postage prices cents PSR Real postage prices cents PCM Money computed postage prices cents PCR Real computed postage prices cents GNPM Money GNP x 109 dollars GNPMC Per capita money GNP dollars GNPR Real GNP x 109 dollars GNPRC Per capita real GNP x 102 dollars DIM Money disposable income (DI) x 109 dollars DIMC Per capita money DI H O ro dollars DIR Real DI x 109 dollars DIRC Real per capita DI x 102 dollars QR Volume of class I mail C T \ O H * pieces QRC Volume of class I mail per ca- pita pieces 80 DEMAND FOR CLASS I MAIL Group I: Demand in money and real GNP terms. P.01(2,34) = 5.29 F>05(2,34) = 3-28 Q = 17.07 - 2.364 PSM + 0.04853 GNPM (.3930) (.0058) R = .9939 R2 = .9879 F = 1390.0 Q = 17.25 - 2.934 PCM + 0.05438 GNPM (.4673) (.0238) R = .9942 R2 = .9885 F = 1455.0 Q = 5.069 - 0.4270 PSR + 0.04594 GNPR (.1772) (.00137) R = .9899 R2 = .9798 F = 824.4 Q = 4.367 - 0.3955 PCR + 0.04693 GNPR (.2026) (.00126) R = .9893 R2 = 9788 F = 783.2 81 DEMAND FOR CLASS I MAIL Group 2: Per capita demand in money and real GNP terms. F.01(2’3i° = 5,29 F.05(2>34) = 3*28 QIC = 114.3 - 11.14 PSM + 0.04039 GNPMC (1.805) (.00155) R = .9880 R2 = .9762 F = 697.0 QIC = 111.4 - 12.99 PCM + 0.04541 GNPMC (1.875) (.00204) R = .9895 R2 = .9791 F = 795-4 QIC = 49.81 - 3.265 PSR + 4.127 GNPRC (1.196) (.2345) R = .9732 R2 = .9471 F = 304.4 QIC = 37.45 - 2.393 PCR + 4.360 GNPRC (1.373) (.2162) R = .9699 R2 = .9408 F = 270.1 82 DEMAND FOR CLASS I MAIL Group 3: Demand in money and real DI terms. F.01(2s3iO = 5.29 F<05(2,3i i) = 3.28 QI = 7-102 - 0.7646 PSR + 0.06790 DIR (.1406) (.00165) R = .9932 R2 = .9865 F = 1090.0 QI = 6.408 - 0.7930 PCR + 0.07033 DIR (.1698) (.00160) R = .9923 R2 = -9846 F = 1244.0 QI = 16.23 - 2.332 PSM + 0.07202 DIM (.3655) (.00218) R = .9947 R2 = .9895 F = 597-0 QI = 16.29 - 2.878 PCM + 0.08046 DIM (.4322) (.00327) R = .9950 R2 = .9900 F = 1675.0 83 DEMAND FOR CLASS I MAIL Group 4: Per capita demand in money and real terms. F.01(2j3l° = 5’29 P.05(2j3i,) = 3,28 QIC = 54.28 - 4.808 PSR + 6.308 DIRC (.9776) (.3034) R = .9803 R2 = .9610 F = 418.8 QIC = 43.15 - 4.483 PCR + 7.743 DIRC (1.185) (.2954) R = .9762 R2 = .9530 F = 344.8 QIC = 108.8 - 10.91 PSM + .0605 DIMC (1.746) (.00226) R = .9887 R2 = .9775 F = 738.4 QIC = 105.2 - 12.60 PCM + .06770 DIMC (1.815) (.00296) R = .9899 R2 = .9800 F = 833.0 DEMAND FOR CLASS I MAIL 84 Group 5‘ - Demand in money and real prices. Ft01(l,35) = 7.42 F.05 d s35) = 4.12 QI = -3.771 + 8.186 PSM (1.009) R = .8079 R2 = .6527 F = 65.76 QI = -0.7579 + 7.215 PCM (.5897) R = .9003 R2 = .8105 F = 149-7 QI = 46.23 - 3.495 PSR (.8702) R = .5617 R2 = .3155 F = 16.14 QI = 40.93 - 2.604 PCR (1.234) R = .3360 R2 = .1129 F = 4.453 85 DEMAND FOR CLASS I MAIL Group 6: Per capita demand in money and real prices P.01(1’35) = 7'k2 p.05(1j35) = H'12 QIC = 58.37 + 27.50 PSM (4.63^) R = .7082 R2 = .5012 F = 35.2 QIC = 64.96 + 25.20 PCM (2.968) R = .8204 R2 = .6731 F = 72.08 QIC = 252.7 - 16.26 PSR (2.948) R = .6819 R2 = .4650 F = 30.42 QIC = 240.9 - 14.33 PCR (4.400) R = .4823 R2 = .2326 F = 10.61 86 Average Cost Models The next 12 models are presented in three groups of four models each. Group 1 consists of cost models in money terms. Group 2 presents the cost models in GNP deflated real terms. Finally, Group 3 repeats Group 2 in DI deflated real terms. The following nomenclature applies to the average cost models: Table 6 AVERAGE COST MODELS NOMENCLATURE SYMBOL MEANING SCALE/UNITS UCM Unit cost in money terms cents UCRG Unit cost in GNP deflated real terms cents UCRD Unit cost in DI deflated real terms cents Q Quantity g x 10 pieces a2 Quantity squared x 1018 x S - j ^ QC Quantity per capita 2 x 10^ QC2 Quantity per capita x 10^ x S2 NOTE: 2 2 Q was scaled by S.^ and QC by S2 because of the BMD print out limitations. S, = (1/1000), yielding Q2/10 and^QC^ (1/10) and Sp = /1000. AVERAGE COST OF MAIL Group 1: Average cost in money terms. 87 P.0i(2,3^ = 5.29 F>01(1,35) = 7*42 UCM = 5.000 - .1543 Q + (.01716) R = .9810 UCM = -.3305 + .0589 Q (.00515) R = .8880 UCM = 8.556 - .05005 QC (.02173) R = .7873 UCM = -1.275 + .01220 Q (.00195) R = .7263 F.05(2,34) = 3.28 F.o5(1,35) = 4.12 .01859 Q2 x 10"1 .00149) R2 = .9623 F(2,34) R2 = .7886 F(l,35) + .09204 QC2 x 10"3 (.03202) R2 = .6198 F(2,34) R2 = .5274 F(l,35) 434.3 130.5 27.71 39.07 2 Note: The numbers to the right of the Q terms are the scale factors discussed in the previous table. AVERAGE COST OP MAIL 88 Group 2: Average cost in GNP deflated real terms. P.01(2’32° = 5,19 P.05(2’3i° = 3,28 F.01(l*35) = 7.42 p.05(1*35) = 4.12 UCRG = 12.52 - .3042 Q + 0.02677 Q2 x 10-1 (.02056) (.00178) R = .9329 R2 = .8702 P(2,34) = 114.0 UCRG = 4.9000 + .00285 Q (.00722) R = .0666 R2 = .0444 F(l,35) = .1560 UCRG = 19.75 - .09018 QC + .13025 QC2 x 10“3 (.01637) (.02413) R = .6940 R2 = .4817 F(2,34) = 15.80 UCRG = 5.838 - .00210 QC (.00180) R = .1938 R2= .0376 F(1,35) = 1.3636 89 AVERAGE COST OP MAIL Group 3: Average cost in DI deflated real terms. F.01(2,34) = 5.29 P>05(2,34) = 3-28 p.01(1’35) = 7,42 f .05(1j35) = 4,12 UCRD = 12.28 - .3146 Q + 0.02821 Q2 x 10"1 (.02011) (.00174) r = . 91134 R2 = .8900 P(2,34) = 137.6 UCRD = 4.252 + .00897 Q (.00754) R = .1973 R2 = .0389 P(1,35) = 1.417 UCRD = 20.00 - .0952 QC + .1397 QC2 x 10“3 (.0178) (.02626) R = .6757 R2 = .4566 F(2,34) = 14.28 UCRD = 5.063 - .00075 QC (.00194) R = .0653 R2 = -0043 F(l,35) = .1498 90 Demand for Discounted Mail In this part, we present 18 models, divided into six groups of three models each. A group consists of three versions of demand. The objective was to examine whether simpler models could be used instead of the more complex ones. Unfortunately, as can be seen, the models p were either statistically unacceptable, e.g., R = .0056 for group 2, or logically unacceptable, e.g., +1.329 r for group 2, or both +1.563 PDD for group 3- The last two have positive coefficients. For the 18 models of this part, the following nomenclature applies: Table 7 DISCOUNT DEMAND NOMENCLATURE SYMBOL MEANING SCALE/UNIT QD Quantity of mail g x 10 pieces PDM Price in money terms cents PDG Price in GNP deflated real terms cents PDD Price in DI deflated real terms cents QDC Quantity per capita x 10^ pieces r Fraction of payment (r=l-d) percent 2 r r squared x 10“3 DEMAND FOR DISCOUNT MAIL 91 Group 1: Money terms. Ft01(3,33) = 4.44 F.01(2934) = 5.29 F.01(lj35) = 7,42 F.05(3S33) = 2.89 F.05(2j34) = 3,28 F.05(1j35) = 4,12 (1.771) F(3,33) = 67.09 QD = -215.7 + 2.210 PDM + 8.463 r - 7.030(r x 10 (1.463) (1.906) R = .9269 R2 = .8591 QD = -22.08 + 4.437 PDM + 0.9190 r (1.618) (.1789) R = .8899 R2 = .7919 P(2,34) = 64.68 QD = 12.93 + 10.73 PDM (1.389) r = .79/10 R2 = .6304 F(1,35) = 59.70 DEMAND FOR DISCOUNT MAIL 92 Group 2: Real terms (GNP deflated). F.03.(3*33) = 4.44 F.05(3*33) F.01(2*34) = 5*29 f .05(2*34) F.01(1*35) = 7’42 f .05(1*35) QD = -233.8 - 3.876 PDG + 9.590 r - (1.287) (1.572) R = .9390 R2 = .8817 QD = -21.74 - 4.117 PDG + 1.329 r (1.718) (.1206) R = .8846 R2 = .7826 QD = 38.81 - 1.589 PDG (3.589) R = .0746 R2 = .0056 = 2.89 = 3.28 = 4.12 7.895 (rxlO-1)2 (1.500) F(3,33) = 81.97 F(2,34) = 61.19 F(l,35) = -I960 93 DEMAND FOR DISCOUNT MAIL Group 3: Real terms (El deflated) F>01(3,33) = 4.44 F>q5(3,33) = 2.89 f .01(2j34) = 5*29 F.05(2>3i° = 3,28 F>01(1,35) = 7-42 F.05(1*35) = 4.12 QD = -235.8 - 3-577 PDD + 9.565 r - 7.833 (r/10)2 (1.303) (1.600) (1.528) r = .9367 R2 = .8774 F(3,33) = 78.73 QD = -25-24 - 3.932 PDD + 1.374 r (1.718) (.1257) R = .8831 R2 = .7798 F(2,34) = 60.20 QD = 29.25 + 1.563 PDD (3.441) R = .0765 R2 = .0059 F(1»35) = .2063 94 DEMAND FOR DISCOUNT MAIL Group 4: Per capita - money terms. F>01(3,33) = 4.44 F>05(3,33) = 2.89 P 01(2,34) = 5.29 p.05(2’3i,) = 3,28 F.03.(1*35) = 7.42 p .o5(1*35) = 4*12 QDC = -1178. - 18.17 PDM + 49.052 r - 40.70(r/10)2 (8.647) (11.26) (10.46) R = .8534 R2 = .7283 F(3,33) = 29.49 QDC = -57.65 - 5.282 PDM + 5.367 r (9.503) (1.051) R = .7770 R2 = .6037 F(2,34) = 25.89 QDC = 146.8 + 31.47 PDM (8.135) R = .5473 R2 = .2995 F(l,35) = 14.97 DEMAND FOR DISCOUNT MAIL 95 Group 5- Per capita - real terms (GNP deflated). F.01(3»33) = p.05(3’33) = 2.89 F>01(2,34) = 5.29 F>05(2s34) = 3.28 P.01(1,35) = 7.^2 P.05(1,35) = 4.12 QDC = -749.0 - 43.10 PDG + 37-24 r - 30.49(r/10) (4.680) (5.714) (5-451) ,2 2 R = .9551 R = .9137 F(3,33) = 116.4 QDC = 70.30 - 44.03 PDG + 5.337 r (6.427) (.4509) R = .9121 R2 = .8320 F(2,34) = 84.19 QDC = 313.5 - 33.88 PDG (14.21) R = .3739 R2 = .1398 F(1 j'35) = 5-688 DEMAND FOR DISCOUNT MAIL 96 Group 6: Per capita-real terms (DI deflated). F#01(3,33) = 4.44 F.05(3»33) = 2.89 F.oi(2,34) = 5.29 p.05(2j34) = 3,28 P>01(1,35) = 7-42 F.05(1*35) = 4.12 QDC = -762.5 - 41.82 PDD + 36.88 r - 29.67(r/10)2 (4.929) (6.054) (5-779) R = .9503 R2 = .9031 F(3,33) = 102.6 QDC = 34.89 - 43.16 PDD + 5-849 r (6.501) (.4759) R = .9088 R2 = .8258 F(2,34) = 80.60 QDC = 266.8 - 19.78 PDD (14.30) R = .2277 R2 = .0519 F(l,35) = 1.91 97 Evaluation and Summary The pertinent statistical data has been presented with the models for the reader's convenience. Since more than one hypothesis may be consistent with the same set of data, arbitrary criteria must ultimately serve to decide on the acceptance or the rejection of a model. ^ Towards this end, we list three of the more common cri teria below as they apply to various models. (a) Rejection on logical grounds: Although a coefficient may be precise (low standard error), we may still reject it if the sign is contrary to theory. For example, the coefficient of PDM on page 91 is +10.73* with standard error of 1.389* and by most standards rela- 12 tively small. Yet, the "+" sign would make class I mail a Giffen good. The author believes that logic 13 should rule this out. ■^Lawrence Klein, An Introduction to Econometrics (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., l9b2), p. 121. 12 An unfortunate practice in statistics is hereby illustrated. The term, standard error, is the standard deviation of the sampling distribution. But the variance itself has no special name. For simple regression, of course, rules (b) and (c) are equivalent. ^Based on the fact that the income elasticity of demand was positive, in fact, close to one. 98 14 (b) Rejection on statistical ground one: 2 Another reason for the rejection of a model is when R is not believed reliable. An indication of this is given by a low empirical F value. (c) Rejection on statistical ground two:1^ Finally, though a model may pass tests (a) and (b) above, it may still be rejected as useless if the standard error of a coefficient is too large. For example, the coef ficient of PDM on page is -5.282. The sign is right, and F is acceptable at the one percent level of signifi cance, but the standard error is 9-503. The latter would include zero in the confidence interval. In all of the 5^ models, more than half would be rejected on the basis of the above listed criteria. Of the acceptable models, we must chose three; this is the first task of Chapter V. Gerhard Tintner, Econometrics (New York: John Wiley and Sons, Inc., 1965), p. 88. 15 J. Johnston, Econometric Methods (New York: McGraw-Hill Company, 1963)s p. 2T. CHAPTER V APPLICATIONS In this Chapter, we select three models out of the 5^ fitted equations. The empirical coefficients are then applied to the algebraic models of Chapter III. These equations were programmed with income and the dis count rate as parameters. All 52 cases (four values of d and 13 values of income) were processed or simulated. Finally, the results of the simulation were analyzed and evaluated. Chapter V is organized in three major sections. The first section discusses the selection of the final models. In addition, relevant microeconomic properties are also developed. In the second section, we delineate how the simulation was effected or mechanized. The re sults and some brief remarks close this section. Finally in the third section, we present the findings of this project. 99 100 Selection of the Final Models In order to render the theoretical structure of Chapter III operational, we must select one regular demand model, one cost curve and one discount demand model. The properties of the selected models are presented in this section which is organized in four parts. The first part presents the model for the regular demand. The second presents the discount market demand. In the third ! part, the selected average cost curve is presented. Fur thermore, for each of the selected models, the Durbin-Wat- son statistic d (formerly the Von Neumann ratio) for a one sided test for positive autocorrelation was computed and shown. In addition, the upper and lower bounds dy and d^ for the critical values of d at the one percent level of significance were also shown. The interpretation of the 2 statistic is as follows. If dv < d^ reject the hypothesis that the series of residuals is random. Conversely, if d > dy do not reject the hypothesis. For d values between these bounds, the test is inconclusive. In the last part, 1Ideally, we would have liked to select the models with the best statistical properties, but this was not practical. For example, the best cost curve is in money terms (see page 87). 2 J. Johnston, Econometric Methods (New York: Mc Graw-Hill Company, 1963), p. 192. 101 a brief summary concludes this section. Regular Demand Qr The demand curve selected for our objective is Qr = 7-102 - 0.7646 Pr + 0.0679 Y (1) where Y is the real disposable income. Qr and are quantity and price, respectively. The inverse demand function is: Pr = (9.289 - 0.08880 Y) - 1.308 Qr (2) Multiplying equation (2) by and differentiating with respect to Qr yields marginal revenue, MR^. MRr = (9.289 - 0.08880 Y) - 2.616 Qr (3) Figure 12 illustrates these curves evaluated at Y = 600 Q x 10 dollars. Unfortunately, neither demand functions had acceptable models in money terms. Hence, the selected models are a compromise. From equation (1), we have: < 5 Q , 6Y = 0.0679 (4) Hence, our commodity is a normal good at the current level of real disposable income. Also, the income elasticity of demand is: 102 6Qr . Y 0.0679 Y (5) 6Y Q. 'r Q. r At Y = 600 x 109 and Qr = 45 x 109, we have 3 Hence, we do not quite have a luxury good. Furthermore for the data of this model the empirical d statistic was computed at 0.5925* For the structure of this model however, d^ is equal to 1.17. Hence rejection of the random series hypothesis is in order. Still, the model was kept for the following reasons. Ordinary least squares point estimates are unbiased, despite the presence of positive autocorrelation. This follows from the fact that the expressions for the expected values do not contain Where X is the matrix of independent variables and Y is the dependent vector, the equation does not contain the deviations. In addition, the application of the U.S. Post ^This, however, is rather arbitrary since income is a ceteris paribus clause. For example, at Y = 650 x Econometrics (New York: John Wiley and Sons, Inc., 1970), pp. 138, 332. 4 the deviation terms. Specifically we have 3=(X'X) 1X’Y. 109, eyQr > i. ^Ronald J. Wonnacott and Thomas H. Wonnacott 103 Office data to our theoretical models was undertaken for exposition purposes only. Hence it was felt that the problem of the understated variances of the OLS coeffi cients which results from autocorrelation was beyond our scope. Discount Market Demand The selected model is: Qd = -235.8 - 3-577 Pd + 9-565 r - 7-834 (r/10)2 (7) Letting r = 1 - d and expanding, yields Qd = -62.6 + 6.105 d - 0.07833 d2 - 3-577 Pd (8) Differentiating with respect to d yields: <5Qj = 6.105 - 0.1567 d (9) Hence, for d< (6.105/0.1567) = 38.96$, an increase in d shifts the demand to the right. Above d = 38.96$, the converse takes place. In addition, the elasticity of demand with respect to the discount rate is: e / a = ^ (6.105 a - 0.1567 a2) (10) a Qa Again, this is a parabola concave to the Qd axis with a maximum value at d = 38.96. From equation (8), 5 we obtain the average revenue function. Pd = (-17.50 + 1.710 d - 0.0219 d2) - 0.2796 Qd (11) Proceeding as in the previous section on "Regular Demand Qrs" the marginal revenue, MRd, is: MRd = (-17.50 + 1.710 d - 0.0219 d2) - 0.5592 (12) Finally, the empirical d statistics were computed at 1.48. The lower and upper bounds for the critical value are 1.11 and 1.45 respectively. Hence, the random series hypothesis is not rejected. These curves are illustrated in Figure 11. ^The mathematically valid range of inference for d is obtained by setting P and Q. to zero in equation (11). Solving for d we have: 32.20. d.< 88.85 or = 65.80 > r > 12.15 This, however, is larger than the range of the data over which the fit was made. Specifically, the empirical ranges are: 34.71. ^ d ^ 61.12 or 65.29 > r > 39.88 Of the two d ranges, the statistical one must prevail. This follows from the fact that "...the variance of the prediction increases the further the X0 (here d) value lies from the mean of the sample values employed to com pute a and 3." In fact, the variance increases with the square of the deviation. See Johnston, p. 37- Reference to this difficulty will be made in the following section on 'Simulation." Average Cost Function 105 The selected model is: AC = 12.28 - 0.3146 Q + 0.2821 x 10"2 Q2 (13) where Q = Qr + Qd* The signs of the coefficient indicate a U-shaped cost curve.^ Multiplying by Q yields total costs, TC: TC = 12.28 - 0.3146 Q2 + 0.2821 x 10“2 Q3 (14) Differentiating with respect to Q yields marginal costs as follows: MC = 12.28 - 0.6292 Q + 0.8463 x 10"2 Q2 (15) The lowest point of the average cost curve is: n - .0-3146 = _ Q1 2(o!2821) 55-76 with = 3.2. Similarly, for the marginal curve: / r This is in agreement with Morton S. Baratz, "Cost Behavior and Pricing Policy in the Post Office," Land Economics, XXXVIII (November, 1962), 310 (notes 10, 11, 12, 13). 106 Q = ------ 0 • I2. ?. . 2 --- = 37.2 d 2(0.8463)(10 ) with P2 = 0.7. Again, the empirical d statistic was calculated at 1.41. The upper bound for the critical value is I.38. Therefore, the random series hypothesis is also not rejected. These curves are shown in Figure 11. Summary Three points should be noted. First, the curves are functions of the parameters, d and Y. Both are given to our system. Any normative question must provide its own d and Y. This is further limited in point three below. Second, the relatively flat appearance of the cost g curve is deceptive. In going from Q = 10 x 10^ to g Q = 50 x 10 , the per unit cost is halved, from approxi mately six cents to three cents in real terms. These 2 economies of scale are significant in light of an R = 0.9434 and an empirical F = 137.6. Yet the real price from class I mail went from approximately nine cents to six cents. And lastly, the method by which the discount market was developed in this study renders difficult policy recommendations in II, III and IV class mail. The Figure 11 THE EMPIRICAL MODELS at Y = 600 xlO AC = 12.28 - 0.3146 0.2821 x MC = 12.28 = 0.6292 0.8463 x AC AC MC ----- ' o 0=47.84 x 1$^ 0= 23.41 x 10 Q=23-92 x 10 108 data of class I mall, on the other hand, has been practi cally unmassaged. Hence, inferences would rest on more solid grounds. Simulation In this section, we set the coefficients of the theoretical models of Chapter III equal to the values of the empirical coefficients just selected. This section is divided into three parts. First, a description of how the simulation was carried out is provided. Second, graphs of the more important results are presented. Lastly, we give an evaluation and summary for this section. Mechanization The dependence of regular mail on Y and of dis count mail on d made it necessary to simulate the model for sets of parametric values, (Y,d). For income, we selected 13 values ranging from $100 billion to $700 bil- 7 lion in increments of $50 billion. Again, the discount rate d is given to this study. Still, since d enters the discount demand, we set ^Although this figure might not be reached for another two to three years, we still examine these values. 109 d± = 0.6, 0.5, 0.4, 0.3. That is tosay, we choseequal increments rather than the more correct d = 0.6112, 0.5, 0.4, 0.3471. This was done for ease in computing. Thus, we have 4 x 13 = 52 cases. For each of the 52 pairs, (Y,d), we computed equilibrium constrained values for the prices, quantities, and profits in each market. These figures are shown in the next part on "Results of Simulation," and in the Appendix. In addition, for each of the 13 regular demand curves, we computed the unconstrained price, P , as defined on page 62. Finally, we computed for each case a1 « 1 - fr . pa Thus, d' is the discount the regular consumer obtains as a-result of the discount market, e.g., his O gain in welfare. These values are the subject matter O The choice of d1 as a measure of welfare in lieu of the gain in consumer surplus is based on the fact that both d and d’ are discount percentages, i.e., scalars, and can thus be compared directly. Consumer surplus, on the other hand, has for units, dollar - piece, which is difficult to compare to a 50 percent cut in price for instance. 110 of the following section on "Findings." Results of the Simulation This section contains the graphs and tables of the simulation. Since the graphs are provided for visual aid only, a representative (close to the mean) value of d = 0.5 was selected. The table below each graph lists Q the output for the other values of d. For the same reason, the marginal revenues and cost curves have been left out. Furthermore, since income does not enter the discount function, the curve is the same in each income case. Table 8 lists the intercepts for Pd = f(d) - 0.2796 Qd (16) as a function of d. Only three of the more relevant graphs are pre sented here; the remainder of the graphs are given in the Appendix. ^The output was further suppressed if P < 0 or Pd < 0 or Qr < 0 or Qd < 0. Ill Table 8 DISCOUNT DEMAND PARAMETERS r d f (d) f(d)/0.2796 0.6 0.4 6.070 21.71 0.5 0.5 13.09 46.82 0.4 0.6 15.73 56.27 0.3 0.7 13.99 50.04 In each market equilibrium values are indicated by two arrows at right angles. For example, in Table 99 equilibrium price and quantity in the discount market are 21.94 and 31.07 respectively. P = 21.94 “<" r Qr = 31-07 The starred figures are the unconstrained equili brium values discussed in the previous part on "Mechani zation." n± stands for profits where the subscript indi cates which market, n, unsubscripted, equals total profits. We now turn to the output. Figure 12 DISPOSABLE INCOME = 600 x 109 DOLLARS Discount Market Regular Market 62.57 - 0.2796 Q, 32.42* 13.08 31.07 Q . r 112 Table 9 DISPOSABLE INCOME = 600 x 109 DOLLARS r d P r Pd Q r Qd n r nd n 0.5 0.5 21.94 10.97 31.07 7.58 547.0 50.32 597.3 0.4 0.6 27-16 10.86 27.07 17-41 630.7 121.8 752.5 0.3 0.7 29.69 8.908 25.14 18.19 647.3 90.28 737-5 Figure 13 DISPOSABLE INCOME = 650 x 109 DOLLARS Discount Market Regular Market 67.01 - 1.308 Q. 51.24 I 411 Table 10 DISPOSABLE INCOME = 650 x 109 DOLLARS r d P r Pd Qr Qd n r nd n 0.5 0.5 22.98 11.49 33.66 5.721 630.1 41.34 671.4 0.4 0.6 28.48 11.39 29.46 15.52 726.1 117.3 843-3 0.3 0.7 31.29 9-390 27.31 16.47 747.7 90.17 837.9 115 Figure 14 DISPOSABLE INCOME = 700 x 109 DOLLARS Discount Market Regular Market 12.02 116 Table 11 DISPOSABLE INCOME = 700 x 109 DOLLARS x r d P r Pd Q r Qd n r nd n 0.5 0.5 24.03 12.02 36.26 3.847 719.0 30.07 749.1 0.4 0.6 29.81 11.92 31.84 13.62 827.9 110.5 938.4 0.3 0.7 32.90 9.870 29.48 14.75 855-3 88.27 943.6 117 118 Remarks The first remark refers to the usage of the term "simulation" in this section. The second remark refers to the results of the simulation as they apply to foot note 4 of Chapter I. The inclusion of exogenous variables in econome tric models is an attempt to close an otherwise open system.10 Consequently, the solutions of such models give values of the endogenous variables for values of the exo genous variables. The process of solving such a system over time or over the exogenous variables as they vary over time is known as a "simulation.1,11 Naylor defines a simulation as: Simulation is a numerical technique for conducting experiments on a digital computer, which involves certain types of mathematical and logical models that describe the behavior of a business or economic system (or some com ponent thereof) over extended periods of real time.12 Emile Grunberg, "The Meaning of Scope and Exter nal Boundaries of Economics," The Structure of Economic Science, ed. Sherman Krupp (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1966), pp. 148-165- 11P. J. Kiviat, Digital Computer Simulation: Modeling Concepts, A RAND Corporation Memorandum RM-5378- PR,(Santa Monica, California: August, 1967), P» 4; Robert C. Meier, et al., Simulation in Business and Economics (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1969), PP. 2, 3, 119, 120-123. 12 Thomas H. Naylor, Computer Simulation Techniques (New York: John Wiley & Sons, Inc., 1969), p. 3. 119 This is in agreement with Orcutt's earlier definition: Simulation is a general approach to the study and use of models. As such it furnishes an alternative approach to that offered by conventional mathematical tech niques. In using conventional mathema tical techniques to solve a model the ob jective is to determine, deductively and with generality, the way in which the model implicitly relates endogenous varia bles to initial conditions, parameters, and time paths of exogenous variables. Simulation techniques also are used to solve models, but in any single simu lation run the solution obtained is highly specific. Given completely specified ini tial conditions, parameters, and exogenous variables, a single simulation run yields only a single set of time paths of the endogenous variables. To determine how the behavior of the endogenous variables is more generally dependent on initial conditions, parameters, and exogenous variables may require a very large num ber of simulation runs; and even then induction from specific results to general solutions will be required.^3 Furthermore, an error free simulation is referred to as "deterministic." When stochastic elements enter 14 the simulation, we have a stochastic simulation. Again, this is in line with Shubik's distinction: lo Guy H. Orcutt, "Simulation of Economic Systems," The American Economic Review, L, No. 5 (December, I960), 693-907. 14 Rosser Nelson, "Systems, Models, and Simula tion," Unpublished course outline at the University of California, Los Angeles, p. 5* 120 The variety of problems explored by means of simulation has called for the development of specialized techniques. Most simulations employ only a high-speed digi tal or analogue computer. Those of interest to economists belong almost completely to the former type. There is, however, impor tant work involving simulation utilizing both men and computing machines. Further more, problems in physics and engineering have utilized a statistical device known as the "Monte Carlo Method." There is a highly specialized technical literature on this .... As this is not of prime concern to the economist no further discus sion of the Monte Carlo Method is given here.1^ Thus, in light of these definitions, we feel the word simulation is applicable to our results. The second remark refers to the break-even characteristics of our model. In Figure 11, the average cost curve cuts the regular demand at approximately 3-7 cents per letter. Over the current range of the disposable income (600 - 700 billion dollars), the average cost curve is extremely flat. Thus, an expected regulated price per letter of four cents seems reasonable. In the consolidated market, at a 50 percent discount rate, the price is also four cents per equivalent letter. Since the discount demand (consolidated mail) is a function of the discount rate, ■^Martin Shubik, "Simulation of the Industry and the Firm," The American Economic Review, L, No. 5 (Decem ber, I960), 910-911. no inferences in that market will be attempted. In the case of Class I mail, our findings are the opposite of those of ex-Postmaster General Day. Specifically, he had warned Congress that a break-even corporation would 1 6 double the rate on Class I mail. Our findings, on the contrary, indicate that the price would go from six cents to four cents instead of six cents to 12 cents. Regarding this apparent discrepancy, it should be noted, however, that since 1968, Mr. Day has been retained as counsel by the "Associated Third Class Mail Usersand the latter pays a rate which only covers 73 percent of the j 18 ^ cost. In a similar fashion, Reader’s Digest, a heavy Class II user, also objects to changes in the Class II structure.^ Again, the price paid only covers 26 percent 20 of the cost. Also, our findings are not in accord with the President's (Mr. Nixon) recommendation that first class postage should be raised to seven cents. Rather, we ■^"How a Post Office 'Corporation' Would Work," Changing Times, XXIV, No. 2 (February, 1970), 17-20(editog- iai;. Ibid., p. 19. 1 O J. George Butler, "Toward Postal Reorganization," Christian Century, LXXXVII, No. 4 (January 28, 1970), 107. ■^David Sanford, "Post Office Blues," New Republic, CEXII, No. 12 (March 21, 1970), 20. 20Butler, p. 106. 21 show a recommended drop to four cents. 122 Findings This section concludes this chapter. In essence, it summarizes the findings of this project. Thus, it also concludes this dissertation. This section is or ganized into four parts. The first contains a graphical and tabular description of d’. The second stems from an observation made from the output. Thirdly, this sec tion examines how d' varies with income, holding d con stant. Finally, we have income constant and vary d. This last point is basically what we have set out to accomplish. The Discount Rate d* Table 12 presents values of d' for pairs of income - discount rate values (Y,d), as indicated in the simula tion section. Figure 15 presents Table 12 in graphical form. Controlling Monopoly Referring back to Table , we note that total profits go from n = -112.4 to n = 47-27 as d goes from .4 to .5- Hence, there may exist a value for d such 21 Ibid. 123 Table 12 DISCOUNT TO THE REGULAR MARKET DUE TO THE CONSTRAINT: d' = (1 - P /Pj r a Disposable^ Income x 10y Dollars d = 0.4 d = 0.5 d = 0.6 d = 0.7 100. 0.85 ‘ 0.16 0.01 0.04 150. 0.76 0.19 0.03 0.04 200. 0.69 0.21 0.07 0.04 250. 0.63 0.23 0.08 0.04 300. 0.57 0.25 0.10 0.05 350. 3i 0.26 0.11 0.05 • o o ■ = r 0.28 0.13 0.06 450. 0.29 0.13 0.06 • o o L T V o on • o 0.14 0.07 550. i —i on o 0.15 0.08 600. 0.32 .0.16 0.08 650. 0.33 0.17 0.09 700. 3i 0.34 0.18 0.10 NOTE: Two significant figures should suffice since the "unconstrained price" was a theoretical construct to begin with. * Figure 15 REAL DISPOSABLE INCOME x 109 DOLLARS 125 2 2 that n = 0, that value being between 0.5 and 0.6. Thus, under our assumptions, monopoly profits can be re duced to zero via the price discriminating discount rate d. Figure 16 illustrates the idea using DI = 100 x 10^ dollars. And as in the d'-d graph on page 3» the smaller empirical range restricts the curve to the values shown. Income Varies, d is Constant Referring back to Figure 15, we observe that d' = h(d,Y) where d’ is the unconstrained discount d is the discriminating discount Y is real income, and 23 h is a function. Holding d constant, we see that d1 varies with Y. In fact, the elasticity of d’ with respect to Y for d =0.5 and 600 ^ Y =< . 700 equals: AdJ_ . _Y _ 0.020 650 _ n ,Qii AY d* 100 x 0.330 u>:5y4' 22 The exact value can be obtained from equation (20) in Chapter III. 23 Though this function can be derived, it is cumbersome. For our purpose, the fact of its existence is sufficient. Figure 16 PROFITS VERSUS DISCOUNT 126 n +100 +50 -50 -100 127 Thus, at a constant discriminating discount rate d, a one percent increase in income (for our normal good) yields approximately a 0.4 percent increase in welfare. Income Constant, d Varies Prom Table 9, we plot d’ against d for Y = 600 x 109 dollars. Figure 17 d VERSUS d' 128 We observe that the two discount rates vary in versely with each other. Also, we compute the elasticity of d' with respect to d as follows: Ad1 . d_ _ 0.32 - 0.16 0.55 _ , <r7 Ad d' 0.1 0.24 51 for the upper section of the curve, and — * — = 4 35 Ad d' for the lower section. Thus, for our particular system, a one percent decrease in the discriminating discount rate results in the lowering of the price in the regular market by three to four percent. At this point the reader must be warned that these results are to be interpreted only in light of the assumptions presented in Chapter III as well as the empirical models and their respective restrictions. CHAPTER VI CONCLUSION Summary In this dissertation, we have examined the effects of a price constraint on a discriminator operating in two markets. The constraint was of the form, = (l-d)P2 s where 0 ^ d jl. That is to say, one of the markets was favored by obtaining goods with a policy-set discount rate d. Under the assumptions of Chapter III, and for our specific empirical models, the net effect of the discount was to lower prices of the non-favored or regu lar market.1 This price reduction was coupled with a corresponding loss in profits. However, since our empiri cal firm was operating under decreasing costs, these losses were partly offset by smaller per unit costs under the expanded output. Furthermore, the discount to the relative market d’ was seen to vary inversely with d for each level of disposable income. Finally, we observed 1Relative to our arbitrary unconstrained price. 129 130 that since profits varied with d the latter could con ceivably be used as a policy variable to control monopoly profits, although in this case, the effect on the prices might not necessarily be the best from a consumer stand point . Evaluation From a theoretical aspect, the mathematical de velopment of the analysis proceeded, with one exception, in a routine fashion. Possibly a similar model has been developed elsewhere. A criticism could be addressed, however, at the method employed in unconstraining the price discriminator, namely, by the elimination of the discount market. In a similar manner, the traditional setting of average revenue equal to marginal costs (the competitive price and quantity) is also subject to cri ticism as it avoids the question of how the demand curve : became downward sloping. Still, we felt our method jus- ! 2 ! tified since we had assumed that the discount rate created i the lower priced market. In the empirical work, a second criticism could p ‘ Hence, the elimination of the discount rate was tantamount to the elimination of the discount market. This point was presented in Chapter III, page 61. 131 be leveled at the consolidation of Classes II, III and IV mail into the discount market. There are two parts to this criticism. First, the very idea of consolidation. Second, the validity of the assumption that the difference in the classes is strictly shown by the differences in their costs. Towards the first objection, we offer the follow ing three explanations. First, the Post Office is not a price discriminating monopolist anyway. We simply sti pulated what if it became one. Second, from the very be ginning, our analysis has been directed at the regular market, and not the discount one. Finally, had we not consolidated, we could not have applied the Post Office to our theoretical structure. Namely, if Classes II, III and IV are different products, then we have a multi ple product "firm" but not discrimination. Hence, the consolidation enabled the application of our theory to the empirical world. Regarding the second objection, the method of consolidation, we felt that in light of the limited, but precise data, the simplest assumption was the best. 132 Recommendations for Further Research There are several areas where further research could be launched. Naturally, the first project should be to expand our theoretical model to a multiple-product - multiple- markets price discriminator. That is to say, for each product, the monopolist has a set of discounts. This yields a discount matrix, D, with elements d.., where i stands for the product and j stands for the market. For constraints, we would have: Pd = di1Pr . a11 i’J ij J id Although mathematically, this model could be developed, it is questionable whether a fit could be established in light of the difficulties encountered by this effort. Another indicated study could be the application of our model to the police and fire service rendered to a city. Specifically, we would consider the regular mar ket to be made up of the non-home owners in the city. The discount market will consist of home owners, other property owners, and business men. There are three obvious objections. First, the computation of d would be a major problem. In addition, the profit maximizing motive should be replaced by the more realistic unit cost 133 o minimizing assumption. Finally, there is the more serious objection that this particular study might not be another exercise in mathematical gymnastics. The third and last indicated area for further research would be to develop our model under the general constraint P^ = f(Pr), where f is some not necessarily algebraic function. This would permit the treatment of complicated tariffs and agreements as constraints and thus enable the application of the economic models of third degree price discrimination to international trade. ^Karl Henrik Borch, The Economics of Uncertainty (Princeton, New Jersey: Princeton University Press, 1968), p. 170.______________________________________________________ APPENDIX 134 Figure 18 DISPOSABLE INCOME = 100 x 109 DOLLARS Discount Market Regular Market P 13.08. 13.89 135 | Table 13 9 DISPOSABLE INCOME = 100 x 10 DOLLARS r d Pr Pd Q r «d nd n 0.6 0.4 2.149 1.289 12.25 17-10 -40.76 -71.61 -112.4 0.5 0.5 12.03 6.015 4.695 25-31 31.21 16.06 47.27 0.4 0.6 14.12 5.649 3.095 36.07 30.44 49.14 79.58 0.3 0.7 13.81 4.143 3-332 35.23 31.55 -7.035 24.51 < jO ON Figure 19 DISPOSABLE INCOME = 150 x 109 DOLLARS Discount Market Regular Market 22.61 - 1.308 Q. 13. P = 12.97 r=. 5 17.29 22.63 Q, Q . 137 . Table 14 DISPOSABLE INCOME =150 x 109 DOLLARS r d P P, Q n n r d r d r d 0.6 0.4 3.798 2.279 14.38 13.56 -27.24 -46.28 -73.51 0.5 0.5 12.97 6.483 7-374 23.64 56.99 29.44 86.43 0.4 0.6 15.41 6.164 5.505 34.22 61.52 • 66.05 127.6 0.3 0.7 15-39 4.616 5.523 33.54 61.25 10.75 72.01 i —■ uo CO Figure 20 DISPOSABLE INCOME = 200 x 109 DOLLARS Regular Market Discount Market 27.05 - 1.309 Q. 17.62 7.213 20.68 139 Table 15 DISPOSABLE INCOME = 200 x 109 DOLLARS r d Pr pa Q n r nd n ! 0.6 0.-4 5.486 3-291 16.49 9.939 -7.431 -26.29 -33-72 ! 0.5 0.5 13.92 6.958 10.04 21.93 88.48 40.66 129.1 0.4 0.6 16.70 6.680 7-912 32.38 99-04 80.82 179-9 0.3 0.7 16.96 5.089 7-712 31.85 98.05 26.75 124.8 OtfT Figure 21 DISPOSABLE INCOME = 250 x 10* DOLLARS Regular Market Discount Market 31.49 - 1.308 Q. f(r) - 0.2796 Q 14.88 13.08 46.82 24.08 9-301 Q . 12.70 Q, 20.21 Q, 0 I Table 16 DISPOSABLE INCOME = 250 x 109 DOLLARS r d P r pd Q r n r nd n 0.6 0.4 7.222 4.333 18.56 6.213 18.61 -11.72 6.898 0.5 0.5 14.88 7.442 12.70 20.21 125.7 46.96 175.6 0.4 0.6 18.00 7.200 10.32 30.52 143.0 93.45 236.4 0.3 0.7 18.54 3.563 9.898 30.15 141.9 40.95 182.9 142 Figure 22 DISPOSABLE INCOME = 300 x 109 DOLLARS Discount Market Regular Market 35.93 - 1.308 Q. 13.0 11.36 Q. i —1 -Cr UO Table 17 DISPOSABLE INCOME = 300 x 109 DOLLARS r d P r pa Q r n r "d n 0.6 0.4 9.023 5. 4l4 20.57 2.35 50.85 -2.67 48.18 0.5 0.5 15-86 7.931 15-34 18.46 168.7 56.51 225.2 0.4 0.6 19-30 7.718 12.72 28.66 193.4 103-9 297.3 0.3 0.7 20.13 6.038 12.08 28.45 192.9 53-37 246.3 i —1 J=" - t = ~ Figure 23 DISPOSABLE INCOME = 350 x 109 DOLLARS 40.37 Discount Market Regular Market 0.2796 Q 40.37 - 1.308Q. 22.87 16.85 13-08 46.82 30.87 13-38 ui Table 18 DISPOSABLE INCOME = 350 x 109 DOLLARS r d P r pa Q V % nr nd n 0.5 0.5 16.85 8.426 17.98 16.69 217-4 61.10 278.5 0.4 0.6 20.60 8.24 15.12 26.80 250.2 112.3 362.5 0.3 0.7 21.71 6.51 14.26 26.75 250.9 63.99 314.9 9l7l Figure 24 DISPOSABLE INCOME = 400 x 109 DOLLARS Discount Market Regular Market 24.71 46.82 15.37 Q =20.61 Table 19 DISPOSABLE INCOME = 400 x 109 DOLLARS r d P r pa Q r Qd n r nd n 0.5 0.5 17-85 8.926 20.61 14.89 271.8 63.46 335.3 0.4 0.6 21.91 8.762 17.51 24.93 313.4 118.5 431.9 0.3 0.7 23-30 6.991 16.44 25.04 316.1 72.83 388.9 Figure 25 DISPOSABLE INCOME = 450 x 109 DOLLARS Regular Market Discount Market P =i!9-25 -1.308 Q. P,=f(r)-0.2796 Q, 26.58 P = 18.86 13.08 17.33 37.66 Q =23.23 Table 20 DISPOSABLE INCOME = 450 x 109 DOLLARS r d P r pd «r n r nd n 0.5 0.5 19-86 9.431 23.23 13-09 332.0 63-57 395.5 0.4 0.6 23.22 9.286 19-91 23.06 383-1 122.5 505-6 0.3 0.7 24.90 7.469 18.62 23.34 388.3 79.87 468.1 150 1 Figure 26 DISPOSABLE INCOME = 500 x 109 DOLLARS Regular Market Discount Market P, = f(r)-0.2796 Q .28.50 13.08 r=. 5 46.82 41.05 151 Table 21 DISPOSABLE INCOME = 500 x 109 DOLLARS r d P r pa Q r Qd H r nd n 0.5 0.5 19.88 9.940 25-85 11.27 397.9 61.43 459.3 0.4 0.6 24.53 9.811 22. 30 21.18 459-2 124.5 583.6 0.3 0.7 26.49 7.948 20.80 21.62 467-5 85.12 552.5 152 1 Figure 27 DISPOSABLE INCOME = 550 x 109 DOLLARS Discount Market Regular Market 58.13 21.17 46.82 153 Table 22 DISPOSABLE INCOME = 550 x 10^ DOLLARS r d P r Pd . Q r Qd n r nd n 0.5 0.5 20.91 10.45 28.46 9.432 ' ' 469.6 57.01 526.6 0.4 0.6 25.84 10. 34 24.69 19.29 541.7 124.2 66 5.9 0.3 0.7 28.09 8.427 22.97 19.91 ■ 553-9 88.60 642.5 H U1 - £ = • r~ BIBLIOGRAPHY Averch, Harvey and Johnson, Leland L. "Behavior of the Firm Under Regulatory Constraints," The American Economic Review, LII, No. 5 (December, 1962), 1052-1069. Baratz, Morton S. "Cost Behavior and Pricing Policy in the Post Office," Land Economics, XXXVIII (Novem ber, 1962), 305—3l^r: Bedford, Norton M. Introduction to Modern Accounting. New York: The Ronald Press, 1968. Blaug, Mark. Economic Theory in Retrospect. Homewood, Illinois: Richard D. Irwin, Inc., 1968. Borch, Karl Henrik. The Economics of Uncertainty. Princeton: Princeton University Press, 1968. Butler, J. George. "Toward Postal Reorganization," Christian Century, LXXXVII, No. 4 (Januarv 28, 1970), 104-108. Clemens, Eli W. "Price Discrimination and the Multiple- Product Firm," The Review of Economic Studies, XIX (1950-51), 1-11. Reprinted in Readings in Industrial Organization and Public Policy. Home wood, Illinois: Richard D. Irwin, Inc., 1958. Economic Indicators. Joint Committee Print. United States Government Printing Office, Washington, D.C., Sep tember, 1970. Economic Indicators. 1967 Supplement. Joint Committee Print, United States Government Printing Office, Washington, D.C., September, 1970. 155 156 i Edgeworth, P. Y. Papers Relating to Political Economy. London: Macmillan & Co.7 Ltd., 195*1. "Federal Workers March to a New Drummer," Business Week, MMCXVII (March 28, 1970), 40-42. Grunberg, Emile. "The Meaning of Scope and External Boundaries of Economics." The Structure of Economic Science. Edited by Sherman Krupp. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1966, pp. 148-165. "How a Post Office ’Corporation' Would Work," Changing Times, XXIV, No. 2 (February, 1970), 17-20(editor- ial), Kiviat, P. J. Digital Computer Simulation Modeling Concepts. A RAND Corporation Memorandum, RM- 5378-PR. Santa Monica, California: August, 1967. Klein, Lawrence. An Introduction to Econometrics. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1962. Kuhn, W. E. The Evolution of Economic Thought. Cincin nati: South-Western Publishing Company, 1963. Johnston, J. Econometric Methods. New York: McGraw- Hill Co., 1963- Leontief, W. W. "The Theory of Limited and Unlimited Discrimination," Quarterly Journal of Economics, LIV (1939-1940), 490-501. Marshall, Alfred. Source Readings in Economic Thought. Edited by Phillip C. Newman, et al. New York: W. W. Norton & Co., Inc., 195^ Meier, Robert C., et al. Simulation in Business and Economics. Englewood Cliffs, New Jersey: Pren tice-Hall, Inc., 1969. Moody’s Industrial Manual. New York: Moody’s Investors Services, Inc., 1968. Morgenstern, Oskar. On the Accuracy of Economic Obser vations . 2nd. ed. Princeton, New Jersey: Princeton University Press, 1963. "A National Humiliation," Nation, CCX, No. 13 (April 6, 1970), 386-388,. (editorial). 157 Naylor, Thomas H. Computer Simulation Techniques. New York: John Wiley & Sons, Inc., 1966. Nelson, Rosser. "Systems, Models and Simulation," unpub lished course outline at the University of Califor nia at Los Angeles. Orcutt, Guy H. "Simulation of Economic Systems," The American Economic Review, L, No. 5 (December, I960), 893-907• Pigou, A. C. The Economics of Welfare. 4th ed. London: Macmillan & Co., Ltd., 1956. Rima, I. H. The Development of Economic Analysis. Home- wood, Illinois: Richard D. Irwin, Inc., 1967• Robinson, Joan. The Economics of Imperfect Competition. London: Macmillan & Co., Ltd., 1964. Sanford, David. "Post Office Blues," New Republic, CLXII, No. 12 (March 21, 1970), 19-22: Scarborough, James B. Numerical Mathematical' Analysis. Baltimore: The Johns Hopkins Press, 1966. Schumpeter, Joseph A. History of Economic Analysis. New York: Oxford University Press, 1954. Scitovsky, Tibor. "Some Consequence of Judging Quality by Price," Review of Economic Studies, XII, No. 2 (1945), 100-105. Seligman, Ben B. Main Currents in Modern Economics. New York: The Free Press of Glencoe, 1963- Shubik, Martin. "Simulation of the Industry and the Firm," The American Economic Review, L, No. 5 (December, I960), 906-919• Standard & Poor's Publishers Corporation Records. New York: Standard & Poor's, Inc., December, 1970. Tintner, Gerhard. Econometrics. New York: John Wiley & Sons, Inc., 1965• _________________. "Some Thoughts About the State of Econometrics," The Structure of Economic Science. Edited by Sherman Krupp. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 158 1966, pp. 114-128. Tintner, Gerhard and Patel, Malvika. "Evaluation of Indian Fertilizer Projects: An Application of Consumer’s and Producer’s Surplus," Journal of Farm Economics, XLVIII, No. 3 (August, 1966), 702-714. United States Post Office. United States Post Office Cost Ascertainment Report. Washington, D.C.: United States Government Printing Office, 1936 through 1969- Watson, Donald S. Price Theory and Its Use. Boston: Houghton Mifflin Co., 1963. Winch, David M. "Consumer's Surplus and the Compensation Principle," The American Economic Review, LV (June, 1965), 395-423. Wonnacott, Ronald J. and Wonnacott, Thomas H. Econometrics. New York: John Wiley & Sons, Inc., 1970.

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Creator Gilon, Paul Raphael (author)

Core Title A Constrained Price Discriminator: An Application To The U.S. Post Office

Degree Doctor of Philosophy

Degree Program Economics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag economics, general,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Tintner, Gerhard (committee chair), Niedercorn, John H. (committee member), White, C. Michael (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-530842

Unique identifier UC11362230

Identifier 7200556.pdf (filename),usctheses-c18-530842 (legacy record id)

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Document Type Dissertation

Rights Gilon, Paul Raphael

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

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Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA