Wicksell and the Problem of the "Missing" Equation

Triggered by a stimulating paper by Bo Sandelin (1980), there has been a debate about Knut Wicksell’s theory of capital and distribution that is known as “Wicksell’s missing equation.” To this debate have contributed, among others, Sandelin (1980, 1982), Takashi Negishi (1982a; 1982b; 1985, chap. 9), Larry Samuelson (1982), and Tom Kompas (1992, chap. 4). The issue under consideration is whether or not Knut Wicksell had put forward a theory of capital and interest that is closed in the sense that the data, or independent variables, from which he started suffice to determine the unknowns, or dependent variables, especially the “natural” rates of wages, rents, and interest. The mentioned authors claim that there is one equation “missing” in Wicksell’s theory and that his formal system of equations is underdetermined. The question then is how to close the system in a way that is faithful to Wicksell. The authors under consideration differ in terms of the closures they suggest.

In this paper I will argue that while the contributions under discussion are valuable because they help to clarify some of Wicksell’s arguments and illustrations, their common premise is dubious: there is no equation missing in Wicksell’s theory. The problem is rather that in the course of his endeavor to develop a coherent long-period supply-and-demand analysis of income distribution, Wicksell became increasingly aware of the fact that his attempt to establish the rate of interest as the “reward for waiting” was confronted with a serious, indeed insurmountable, problem: that of defining the “quantity of capital” independently of the rate of interest. He understood that with heterogeneous capital goods and deprived of Eugen von Böhm-Bawerk’s device of the “average period of production” to aggregate them, the initial endowment of the economy of capital could only be given in value terms. Wicksell saw that this undermined the basic idea underlying neoclassical theory: that there is an analogy between the different factors of production—labor, land, and capital—and their rates of remuneration. Originally put forward by Johann Heinrich von Thünen, that idea had inspired several authors, including Léon Walras, Böhm-Bawerk, and Wicksell himself, to elaborate a theoretical edifice explaining the distribution of income in terms of a single principle: that of the (relative) scarcity of the factors of production. In the course of his work Wicksell became increasingly aware that the idea met with considerable difficulties. In particular, whereas the original factors of production—labor and land—can be measured in their own technical units, the capital endowment of the economy had to be given in terms of a sum of value. His lack of enthusiasm for this option—the only one at his disposal, if the supply-and-demand approach to the long-period theory of income distribution was to be adhered to—therefore reflects a fundamental difficulty of the theory. While this is explicitly or implicitly confirmed by those authors who suggest some alternative closures of the system, the inattentive reader might (wrongly) get the impression that these closures allow one to overcome that theoretical difficulty. In this article I will attempt to draw the attention back to the central problem of Wicksell’s theory and to put the discussion about the missing equation in the perspective of his overall intellectual program.

I will show that Wicksell approached the two problems of capital theory, the remuneration of capital on the one hand and its accumulation on the other, in separate logical stages. In a first stage he took the capital endowment of the economy as given; that is, he treated it as a datum or an independent variable, and determined the natural rate of interest in terms of the relative scarcity of capital. It is in this part of the analysis that he deemed it possible to postulate functional relations of known properties. In the corresponding system of equations the...