¹ Pigou, A. C., Economics of Welfare (4th ed., London, 1950)Google Scholar, chap. 17, and esp. section VIII of appendix III.

² von Stackelberg, H., “Preisdiskrimination bei willkurlicher Teilung des Marktes,” Archiv fur Mathematische Wirtschafts und Sozial Forschung, no. 1, 1933 Google Scholar; translated by Peacock, Alan T. and reprinted as International Economic Papers, no. 8 (London, 1958).Google Scholar

³ Robinson, Joan, Economics of Imperfect Competition (London, 1933), chap. 15.Google Scholar

⁴ Vide: Coase, R. H., “Monopoly Pricing with Interrelated Costs and Demand,” Economica, 02 1946, 278–94Google Scholar; Simkin, C. G. F., “Some Aspects and Generalizations of the Theory of Discrimination,” Review of Economic Studies, 1947–1948, 1–13 Google Scholar; Edwards, E. O., “The Analysis of Output under Discrimination,” Econometrica, 04, 1950, 163–72Google Scholar; Clemens, E. W., “Price Discrimination and the Multiple Product Firm,” Review of Economic Studies, 1951–1952, 1–11 Google Scholar; and Bailey, M. J., “Price and Output Determination by a Firm Selling Related Products,” American Economic Review, 03, 1954, 82–93.Google Scholar

⁵ Economics of Welfare.

⁶ If a monopolist were omniscient, and could prevent resale at no cost to himself, a discriminating seller would presumably charge each purchaser a lump sum for all units bought. The total charge would equal the aggregate reservation prices to that buyer. The monopolist would only include in such a transaction those units for which the purchaser's reservation price was not less than his marginal costs.

⁷ The personal attributes that segregate the buyers may be residence, sex, age, occupation, or income, etc. In practice, market separation is frequently caused by national import restrictions. Transportation costs afford another common basis for price discrimination of this kind.

⁸ Thus the seller may have two prices: $10 and $5. Jones' reservation values for successive units may be $12, $8, and $4: then the first unit is bought at $10 (not $5), the second at $5, and the third unit remains unpurchased. Or, if Smith would pay $15 for a single unit he will be prevented from buying it at the lower $5 price.

⁹ In what follows, it is always assumed that the seller has some monopoly power and knows the reservation prices of actual and potential buyers, and that he can somehow prevent resale at no cost to himself. Finally, the income distribution effects of price denomination are ignored, buyer's reservation prices being given.

¹⁰ The demand curve confronting the discriminating monopolist becomes his marginal revenue schedule. The most profitable output for the seller coincides with that which would result from marginal cost pricing by a purely competitive industry. However, although the output decision is similar in these two instances, the income or welfare result is of course very different: all the buyers' surplus that would have resulted from marginal cost pricing without discrimination becomes part of the perfect discrminator's quasi-rents.

¹¹ Thus the 5th unit sold by die monopolist may be the fourth bought by Jones, the 6th may be the second bought by Smith, and the 7th may be the first bought by Brown, etc.

¹² The average cost to the buyer in this last case is 8.3 cents, but this is irrelevant, because he can and does buy 1, 2, or 3 units.

¹³ Special, because Pigou does not limit each buyer to purchasing one unit in his definition of second-degree discrimination. Practically it is easier to segregate buyers when they can only use one unit (e.g., renting a seat at a theatre or in an airplane.) Thus, because in the real world it is difficult to discriminate among buyers and multiple units simultaneously, the N prices and unit purchase case is significant.

¹⁴ For every point on DD′ that might be a Market 1 price, there is a corresponding point on dC indicating the suboptimum combined output, which is related to a suboptimum Market 2 price and hence to a point on S₂. Thus the logical relation goes D to E to P to C or P ₁ to G to P ₂ to B. If the Market 1 price were A, there should be no Market 2 sales.

¹⁵ It is assumed that the output sold in the second market is supplementary to that sold in the first market, in the sense that, if Q ₁ Q were sold in the second market for example, the prime costs attributable to it would be Q ₁ FGQ.

¹⁶ An important suboptimization is that total revenue be maximized for any given combined output. This occurs when dR ₁ equals P ₂. As explained above, Market 1 can then only add or subtract a unit of sale by respectively subtracting or adding a unit of sale in Market 2, so that P ₂ is a sort of second market dR.

¹⁷ The von Stackelberg proof is in algebra, except for a not very elegant geometric approximation, so Figure 1 may have some pedagogical value.

¹⁸ The “related concavity” analysis of third degree discrimination, originally presented by Mrs.Robinson, Joan (Economics of Imperfect Competition, 200, 201)Google Scholar, was considerably refined in 1950 by Edwards in “Analysis of Output under Discrimination.”

¹⁹ Fortunately, because for each customer dC is the only term that alters in the above formula, the ranking of different customers by inelasticity will not be altered as dC declines.

²⁰ A contrast between this Case III-B-2-b and Case III-A (Pigovian third-degree discrimination) should be stressed. In the present instance buyers are grouped because of similarity in price charged. In the III-A case they are grouped because of some personal similarity such as country of residence.

²¹ Economics of Imperfect Competition, 186.

²² Consider two services, one transportation with cramped seats for $100 and the other transportation with comfort and meals, plus a chance to meet interesting people, for $150. The difference in cost to the transportation company is usually far less than the difference in the price charged for the two services. The people who go first class presumably have reservation prices that, relative to $150, are greater than their reservation prices for tourist class travel relative to $100.

²³ In what follows the specifications of any two closely related products are given. The seller suboptimizes by pricing these predetermined goods to maximize profits. The possibility that a product design change might increase profits is not considered.

²⁴ This is essentially the approach of Bailey (“Price and Output Determination”). The same logic has been folio wea by Coase (“Monopoly Pricing”), except that he considers (a) the best price for X given some price for Y and (b) the best price for Y given some price for X so that one can determine (c) the combination of X and Y prices that is the most profitable. However, inasmuch as total revenue, total cost, output, price, and cost per unit are all related for a single product, this general approach could as well be applied to any one of these five dimensions.

²⁵ Note that third-degree discrimination (Case III-A) would involve a vertical and a horizontal Ȳ _x , assuming the two products are not related on the supply side. Given this proviso, the two goods are substitutes so far as buyers are concerned if they are both negatively inclined, and complements if positively inclined. Along either curve (e.g., ), marginal revenue and marginal cost are everywhere equal (e.g., dR_x =dC_x ) for that product considered to have a variable output.

²⁶ Revenue and cost isoquants have been used to analyse price discrimination before (e.g., Stigler, George J., The Theory of Price (London, 1946), Figure 81)Google Scholar, but only discrimination of the third degree.

²⁷ If these products were complements as regards both demand and supply, X_p and , would both be positively inclined.

²⁸ In short, this would be Case III-A, associated with Mrs. Robinson.

²⁹ This would mean a rectilinear C isoquant.

³⁰ The latter “solution”, as indicated above, is usually associated with von Stackelberg.

³¹ Because , and Ȳ _x . intersect at A.

³² This is because, accepting income distributions, the price of a single good is supposed to indicate its marginal worth to buyers.

³³ And, because the C resource investment is assumed best for profits, these ratios also equal unity.

³⁴ This means that the total revenue isoquant that passes through B′—not shown however—would intersect the C isoquant.

³⁵ I.e., a curve that indicates the minimum alternative mixes of X and Y that would leave each and every household no better or worse off than under the B′ outcome. Such a construct was originated by Scitovsky, Tibor (“A Note on Welfare Economics,” Review of Economic Studies, 02, 1941)Google Scholar, its properties were further considered at a later date by Samuelson, Paul A. (“The Social Indifference Curves,” Quarterly Journal of Economics, 02 1956)CrossRefGoogle Scholar, and more recently it has been resurrected for use again by the author (“Welfare and Trade,” Kyklos, no. 3, 1962).

³⁶ This possible asymmetry is an old difficulty encountered in welfare economics. However, in the case of a single firm in a large economy, ambiguities of this kind seem rather unlikely. At least there is no doubt about the intersection of the Scitovsky curve and the C isoquant at A, because if there is price discairnination, the P_x /P_y ratio cannot equal the dC_x /dC_y , ratio there.

³⁷ It is the intersection point of curve (along which dR _x, and dC _x are always equal) and curve Ȳ _x (along which dR _y and dC_y are always equal).

³⁸ Point B′—as shown in Figure 2—must be on this isoquant.

³⁹ Figure 3 shows two additional isoquants passing through point B. These are labelled Ū _x and Ū _y . For example, throughout Ū _x , P_x , equals dC _x . In other words, given some Y output, the Ū _x , curve indicates the X output that will give the most buyers' surplus (not financial profit).