Input Demand Functions for the Profit-Constrained Sales-Maximizer: Income Effects in the Theory of the Firm

Abstract

Several recent papers (Bear [4], Ferguson [6], Russell [11], Winch [16]) have elaborated upon the accepted analysis of the profit-maximizing firm's demand for inputs (as presented, e.g., in Hicks [8, ch. 7] and Samuelson [12, ch. 4]).2 As one might expect in such a wellexplored field, possible cases have drawn a good deal of attention: when could the demand for an input increase with an increase in its price, or decrease as the firm's output increases (with constant input prices)? Both perfect and imperfect competition in the output market have been considered. After tying up a few loose ends in the theory of the profit-maximizer, this paper extends the analysis to the case of the firm which maximizes sales subject to a minimum-profit constraint. The price-output behaviour of such a firm is familiar from Baumol's work [1, chs. 6 and 7], [2, ch. 13], upon which have followed various models using similar (but usually more general) objective functions, discussing both comparative statics and growth (e.g. [15], [14]). How such a firm would vary its input mix as input prices and output vary seems to have been neglected in this literature, however; so a detailed comparative-statics analysis may be worthwhile. Moreover, we do find interesting differences between the behaviour of the profit-maximizer and that of the profit-constrained sales-maximizer, the most important of which is the appearance of an income effect and the possibility of Giffen inputs, in precise analogy with the theory of consumer behaviour. As a result, the perverse case in the standard theory, while remaining rather odd, now fits into a familiar context; and the relationship between consumer and producer theory appears in a new, perhaps slightly more illuminating light. Finally, we derive relationships between the profit-maximizer's and sales-maximizer's elasticities of demand for inputs. When the present author came upon the results discussed below (in a quite different context-see [10]), he immediately turned to the Foundations and Value and Capital. But Hicks wrote, for example: ... In the case of production, we do not have anything similar to the