The early nineteenth century: building economics through geometry and calculus

In the nineteenth century the old-style, approximate use of mathematics as an illustration gradually lost ground: a notorious example is Nicolas-Franςois Canard, who won an Institut de France contest in 1801 and enjoyed celebrity for some time, but subsequently attracted torrents of criticisms.

Initiated by the work of David Ricardo and the so-called English “classical” school, a large part of the reflection of this time emphasized the differential impact of condi­tions of production on individuals according to their position as workers, capitalists, or landowners. Though most contributions were verbal, some ventured into tackling income distribution and the possible tension between wages and profits with the help of mathematics. In England, the so-called “Whewell group” - see, for example, Whewell (1829-50 [1871]), and Henderson (1996) - of mathematical economists used algebra to represent, cultivate and develop Ricardo’s theory. One of them, John E. Tozer, pro­posed in 1838 a model of the social effects of substitution of capital for labour through the introduction of machinery, which Ricardo had judged “often very injurious to the interests of the class of labourers” (Ricardo 1817-21 [1951]: 388). While apparently criti­cal of Ricardo, Tozer’s equations in fact refine his views by bringing to light the specific conditions under which labourers’ loss occurs (Tubaro 2008).

While these authors primarily used algebra the use of calculus also progressed signifi­cantly. A Prussian landlord, Johann H. von Thunen (1850 [1960]), concocted the concept of marginal productivity and the idea that, at an optimum, the marginal productivity of a factor must equal its remuneration. Not only did he use differential calculus to prove his result, but he derived the very economic notion of marginal productivity from the mathematical concept of partial differential (Tubaro 2006).

The early nineteenth century also saw efforts to move forwards in the understanding of utility. Reflection on this topic had already appeared, with the idea that the problem of political economy and the ultimate purpose of all productive activities, is to satisfy human wants at best. However, it seemed difficult to integrate utility into economic analysis, not least because at first glance it appears as a subjective, qualitative notion devoid of any objective, let alone quantifiable, attribute. Mathematics offered a solution: another German writer, Hermann H. Gossen (1854 [1983]), was the first to apply the idea that, even without actual measurements, formalization can support theory-building at a more abstract level, “to develop the possibilities that may occur in enjoyment and place them in mutual relation” (Gossen 1854 [1983]: 10). He used the mathematical notion of a function to reinterpret utility not as an absolute but as a relative magnitude, varying from one individual to another and for each individual, depending on the available quantity of a good. Gossen considered individual goods and the feelings of satisfaction they yield to the individual, thereby liberating himself from the older postulate of a universal, hierarchical ordering of needs defined with respect to monetary wealth. He also provided graphical methods to determine a maximum of utility, which he inter­preted in a purely normative sense - what people ought to strive for in order to live a satisfactory life - without suggesting that they actually maximize. Gossen’s contribution remained unknown and was only rediscovered towards the end of the century, but in the meantime others independently proposed similar solutions.

Economists’ understanding of the market also made significant progress in this time period. The older, rudimentary formula of price determination, expressed simply as the ratio of demand to supply, was gradually replaced by the idea that demand and supply can be conceptualized as two different schedules and that what matters is their point of intersection, where they are equal. Price-quantity diagrams in which demand decreases and supply increases with price appeared from the 1830s onwards, in particular with Augustin Cournot (1838 [I960]) and Karl H. Rau (1841 [1997]). Verri’s version of the eighteenth-century demand/supply formula, based on the number of buyers and sellers, also intuitively conveyed the idea that in monopoly situations prices are higher and quantities are lower than in cases in which several sellers compete. It was Cournot (1838 [1960]) who first gave an analytical account of this idea, while also pioneering the use of mathematics (including algebra, calculus and diagrams) to guide reasoning. He illus­trated how the sole seller of a good controls the entire market and extracts a monopoly rent; with two or more sellers, decisions are interdependent and the market is in equi­librium when each firm’s output maximizes its profit given the output of others; as the number of sellers rises, the weight of each diminishes until no one exerts any influence on prices, and the quantity produced is such that its market price equals (what would be later called) its marginal cost.

Overall, these attempts had limited resonance, and the use of mathematics remained confined to a small minority. Even Cournot, who was otherwise a well-known scientist, did not enjoy any success and reverted to verbal form in later economics writings (1863). Also, when his 1838 book is later rediscovered and eventually gains the recognition it deserves, it is often believed to be an isolated accomplishment. The other early math­ematical economists remain little known today. Especially the German contribution is insufficiently appreciated, partly shadowed by the Historical School which, in the second half of the nineteenth century, loudly dismissed all efforts to identify universal regulari­ties in the name of empirical and inductive reasoning. Nevertheless, a growing literature on the topic may lead to a more balanced appreciation in future (see, for example, Theocharis 1961 [1983], 1993; Baumol and Goldfeld 1968; Mirowski 1989; Ingrao and Israel 1990; Darnell 1991; Baloglou 1995).