Chapter 9: The Product Exhaustion Theorem
Cassey Lee There are many mathematical theorems that are associated with Leonhard Euler (1707–1783), one of the greatest mathematicians. One in particular found its way into economics, namely Euler’s work on homogeneous function of the first degree, which was applied in the theory of factor pricing and income distribution. In the 1880s, following Ricardo’s theory of rent, Clark (1889, 1891, 1899, 1901), P. Wicksteed and Wicksell argued that any variable factor input is paid its marginal product (Blaug 1997). This led to the formulation of the ‘Product Exhaustion Theorem’ which states that the total product is exhausted if each factor input is paid its marginal product. Wicksteed (1894) provided an early mathematical expression of the theorem but did not relate it to Euler’s work (Stigler 1941, fn.1, p.326). It was A.W. Flux (1894) who pointed out the theorem’s association with Euler’s work. Aside from the importance of the theorem to factor pricing, it is also widely used as a basis for a marginal productivity theory of income distribution. The simplest way to illustrate the theorem is by using a two-factor model. At the core of the theorem is an assumption that factor inputs such as labor (L) and capital (K) are each paid a price (PL and PK, respectively) that is equal to the value of its marginal product (MPL and MPK, respectively). If we assume that the total revenue (PQ # Q) collected from the sale of a product is equal to the total cost of production, we...
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