Marginal variations, limits, and the continuum
The second half of the nineteenth century saw growing interest for utility interpreted as in Gossen: a relative notion, depending on the available quantity of a good. Emphasis was on the distinction between the total amount of utility and “marginal” utility, namely, the change in the level of utility that results from a given increase in the quantity of the good.
With this solution, utility became tractable and gained right of citizenship into economic theory.The importance of thinking in terms of marginal variations rather than total magnitudes proved so useful to account for utility and demand that it was subsequently extended to supply. Notions of marginal productivity and marginal cost had already been sketched, notably by Cournot and Thunen, but were rediscovered independently, refined and generalized in the 1890s by, among others, John B. Clark, Philip H. Wicksteed, and Knut Wicksell. Originally grounded in distinct traditions of thought, models of consumer demand based on utility and models of producer supply based on productivity/cost began to be seen as two symmetric poles, sharing a similar formal structure. On the whole, marginal reasoning seemed so important that the economic thought of this time is often referred to as “marginalism”.
At the beginning, many adopted a discrete framework of analysis and focused on small but finite changes in quantities and the resulting changes in levels of utility or profit. Use of differential calculus initially built directly on this view, taking into account progressively smaller units until they could be said to be “infinitesimally small” - an ambiguous notion that dated back to the Leibnizian origins of differential calculus. A sounder approach saw the light in mathematics towards the mid-nineteenth century, when the notion of limit was forged and calculus was entirely rebuilt on this basis. Economic thought was affected by these results in that emphasis gradually shifted towards the assumption of continuity (also based on the notion of limit) with a continuous representation of the objective function. Eventually, insistence on the effects of marginal increments was replaced by a more straightforward determination of the optimum as the point where the partial derivative is zero.
Though obviously inconsistent with the physical properties of many real-world goods, the continuum appeared as a reasonable approximation of situations in which there is a finite but very large number of elements. The main reason to adopt this approximation was to allow for use of calculus, more powerful and elegant than the finite methods available at the time. Notice, also, that the difference between continuity and differentiability was unknown in mathematics until the late nineteenth century, and the two were taken as equivalent.