The Theory of Cost and Production in the Multi-Product Firm

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.In this paper the theory of cost and production in the multi-product firm is formulated as a problem of minimizing a convex function subject to convex inequality constraints. The Kuhn-Tucker theorem is used to solve the problem. Differences between cost and production theories for the single-product firm and for the multi-product firm are noted briefly. 1. INTRODUCTION THE MULTI-PRODUCT firm occupies a prominent place among important but neglected topics in economic theory. Barring the programming approaches, a search of the literature reveals no comprehensive theoretical treatment of the subject. Indeed the brief development by Hicks in the mathematical appendix to Value and Capital [6] is probably the most thorough treatment in the sense of considering both the production and sales activities of the firm. Since Hicks allots only four pages to this topic, this is a remarkable state of affairs. While the literature on all phases of the multi-product firm is scanty, a few writers have considered the selling side of such a firm; the works of Bailey [1], Clemens [3], and Weldon [10] may be cited in this connection. On the cost and production side the literature is virtually non-existent. Even the work of Carlson [2], a standard reference for twenty years, deals only with the special case of joint cost but not with any other aspect of the multi-product firm.The purpose of this paper is to present a theory of cost and production for the multi-product firm. In developing this theory, the traditional concept of the firm as a social mechanism that connects the markets for factors of production with the markets for finished products will be retained. This is in contrast to the usual "inward looking" view of the firm that appears in the programming literature. But once the problem is clearly stated in traditional terms, it becomes apparent that the Kuhn-Tucker theorem, which can be thought of as the logical basis of the optimization techniques of activity analysis, can be used in a straightforward way to solve the problems of cost and production in the multi-product firm.The solution of the problem will show that the optimum conditions for the multi-product firm are different from those of the single-product firm. These differences are discussed briefly in the final section of the paper.

Podstawy krótkookresowych decyzji produkcyjnych w przedsiębiorstwach wieloasortymentowych

“…Nadal więc pomijane jest, chociaż w mniejszym stopniu niż w omawianej dotychczas metodologii, to, że "firma wieloasortymentowa nie może być traktowana jako zbiór firm jednoproduktowych" [Pfouts 1961, s. 651] 9 . W teorii ekonomii problem wyboru optymalnej struktury produkcji w firmie wieloasortymentowej przedstawił Pfouts [1961]. Problem decyzyjny zdefiniowany w postaci przedstawionych poprzednio pytań: -co produkować?…”

Optimal investment, pricing and replacement of computer resources

“…For decreasing values of zp,, equality is approached as the successive long-run marginal cost values decrease and the associated quasi-rent values increase. Unless equality occurs for some zo,>O, (7) and (8) have no physical meaning, Therefore, the first relation in (7') must become a strict equality for some v and each s, and the sum of the quasi-rents will be strictly positive by Theorem 1. Now, over the time frame v<t< T < m we can substitute from (8) into (7') for the quasi-rents; simplifying, we obtain which is a strict inequality Vu, unless qlrr>O for some t>v. The discounting process guarantees that in the limit as t+=, (28) must become an equality by (7').…”

“…Moreover, since the discounted marginal return from salvage is a decreasing function of time, and the quasi-rents in (8) have been eliminated for all r>t-l, we have (27). That is, whenever a sale is initiated, all of that vintage is sold (or, equivalently, its economic value is "written off" a t time t).…”

Memoriam to Professor Ralph William “Bill” Pfouts

“…He was characteristically patient and giving in working with us; his comments were probing, articulate, and much appreciated. Again he demonstrated humility-he did not point out our failure to include his groundbreaking papers on the theory of multi-product firms (Pfouts 1961(Pfouts , 1964 in our surveys of the literature.…”