Demand Relationships and Pricing Decisions for Related Products zy RENE P. MANES, FRANCOISE SHOUMAKER and PETER A. SILHAN Collegeof~mmerceandwroiwSSA~ ti011, University of IUhk zyxwv at Urbeaa-Chmp- USA zyxw Tbis paper presents m terms of price zyxwvuts changes zyxwvut the theoretical conditiom for opthnnl pricing when products are demand-related. It then mggests an approach whicb could be lLped by companies to provide zyx guidance m the pricing of such products over time. INTRODUCTION Recent papers in this journal by Moms and Night- ingale (1980), hereafter MN, and Hill (1982) have considered problems related to third-degree price discrimination (Robinson, 1933, Book V). Basi- cally, a zyxwvuts firm produces a single product for which different demand functions can be identified, usu- ally in spatially separate markets. It sets prices so as to maximize profit m = px(Qd + Py(Qy) - C(Qx + Qy) (1) (where Qx, Qy and Px, Py are the amounts (and prices) of the single product sold in separate mar- kets) by equating marginal cost of output as a whole (Qx + Q,) with the individual marginal revenues of Qx and zyxwvuts Qy. This result implies that Px(1- 1/qd = Py(1- 1hY) (2) with qX and qy being price elasticities of the de- mands for Qx and Qu, respectively (Henderson and Quandt, 1980). In MN (1980), a firm produces from a single facility one product which is then differentiated as its own proprietary brand and as a retailer brand. For example, a producer of kitchen appliances mar- kets a blender under its proprietary name and also produces and sells a similar item to a major retailer, like Sears, which sells it under its own label. Al- though unit production costs of Qx and Qu are one and the same, the promotional costs of the prop- rietary and retail brand items vary. Consequently, producer profit is = PX(Qx) + PY(Qy) - C(Qx + Qy) - Cx(W - Cy(Qy) (3) where Cx and Cy are differing promotional costs of Qk and Q,. Assuming that aC/aQx = aClaQv, opti- mality conditions now are Although several questions can be raised about the model represented by Eqn (3),’ Hill addressed one in particular (1982) which will concern us here. That is, can the demands for Qx and Q, be as- sumed to be independent? Is it realistic to expect third-degree price discrimination policy to apply to products which are available to the same set of consumers? Subject to the further analysis and qualifications in Brunner’s Comment (1984) and Hill’s Reply (1984), Hill addressed the implications of sub- stitutability and complementarity and derived pric- ing policies for a manufacturing firm’s several pro- ducts which may be competing with or comple- menting each other in the marketplace. Although the market conditions in Hill’s model are not iden- tical to those in MN, Hill follows the MN analysis closely and his results are directly applicable to the proprietary-retail brand situation as well as to other situations in which the separation of markets for a single product or for similar products is not com- plete. THEORETICAL CONDITIONS FOR OPTIMAL PRICING In dealing with models, such as Hill’s, economists usually consider marginal revenues and marginal costs in terms of changes in the level of sales and production quantities. This is the approach taken by Hill (1982) and commented on by Brunner. For purposes of pricing policy, however, we suggest that it may be more useful to state optimality conditions CCC-0 143-6570/84/OOO5-0 120$0 1.50 120 MANAGERIAL AND DECISION ECONOMICS, VOL. 5, NO. 2, 1984 zyxwvut 0 Wiley Heyden Ltd, 1984