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.

A T one time the proposition that an increase in the wage rate would call forth a smaller number of hours of work from an individual laborer was evidently accepted by economists without qualification. For example, Professor Pigou in I920, arguing that the imposition of a tax on a laborer (the opposite of a wage increase) would result in his working more hours, wrote: Since a part of his income is taken away, the last unit of income that is left to him will be desired more urgently than the last unit of income that would have been left to him if there had been no taxation. But the last unit of energy that he devotes to work will not affect him differently from what it did. Consequently, there will be a tendency for him to work a little harder than he would have done otherwise. 1 Again in I928, Professor Pigou put the same arguments more specifically. Then, since income is taken away from the taxpayers, the marginal utility of money to them is raised, but the marginal disutility of work is unchanged. And Professor Pigou continued with a statement on the attainment of an equilibrium position: Hence, unless they are somehow impeded, they will increase the amount of work done, and so of income obtained, up to the point at which the marginal utility of income and the marginal disutility of work done to secure it again become equal-which is obviously the optimum. 2 (Italics in original.) Similarly Professor Knight in I92i argued that if men act rationally, . they will at a higher rate divide their time between wageearning and non-industrial uses in such a way as to earn more money, indeed, but to work fewer hours. 3 (Italics in original.) Both Professor Pigou and Professor Knight based their conclusions on an assumption of diminishing marginal utility of income. The unanimity of expert opinion was broken by Professor Robbins in I930.4 Phrasing his argument in terms of the effort (as distinguished from the money) prices of the goods constituting real income, Professor Robbins pointed out that it is the elasticity of the demand for income with respect to effort that determines the responsiveness of the individual's hours of work to changes in wage rates. If this elasticity is greater than unity, a decrease in wage rates (or the imposition of a tax) will result in fewer hours of labor. If the elasticity is less than unity a decline in wage rates or an increase in taxes will result in more hours of labor.5 Thus Professors Pigou and Knight had implicitly assumed that the elasticity of demand for income in terms of effort is less than unity. Professor Robbins agreed that the deductions of the earlier authors justify the downward slope of a curve representing demand for income in terms of effort prices, but he contended that nothing in their arguments shows that the elasticity of such a curve is necessarily less than unity.6 Professor Paish presented much the same viewpoint as did Professor Robbins, but Professor Paish based his analysis on the demand for leisure, which he treated as a good.7 If income is decreased as a result of an increase in taxation, argued Professor Paish, the result may be either to increase or to decrease a person's demand for leisure. Assuming a flat rate tax increase or a decrease in hourly wages, both aggregate income and marginal income decline. The decline in aggregate income will cause a person to demand fewer hours of leisure. The decrease in marginal income, however, will have the opposite effect; it reduces the cost of addi-