- The Origins of the von Thünen-Mill-Pareto-Wicksell-Cobb-Douglas Function

Scholars have known for some time that the function originally attributed to Charles Cobb and Paul Douglas (see Cobb and Douglas 1928 and Douglas 1934, 1948) had been used by Knut Wicksell in as early as 1896 (Wicksell [1896] 1958).^{1} This was noted by George Stigler (1941, 295), Joseph Schumpeter (1954, 1042), and others. (For references to the later use of the function by Wicksell, see Samuelson 1979.) Chris Weber (1998) has recently shown that the analysis of consumer demand by Vilfredo Pareto (1892) implies a utility function that is Cobb-Douglas in form.

There are, however, at least two other precedents that predate Pareto. Johann von Thünen actually wrote down a form of the Cobb-Douglas production function sometime before his death in 1850 (Dempsey 1960). In elaborating his version of the theory of comparative advantage in the third edition of his *Principles*, John Stuart Mill ([1848] 1852) used a system of demand equations that imply a Cobb-Douglas utility function. The history of von Thünen's accomplishment was recorded by the present writer (Lloyd 1969), and that of Mill was noted by John Chipman (1965, 485), but the contributions of von Thünen and Mill seem to have passed unnoticed by later historians of the function. **[End Page 1]**

This article repeats briefly in section 1 the claims that von Thünen discovered the so-called Cobb-Douglas function and that Mill implicitly used the function. It then asks in section 2 why this function has played such a seminal role in production and utility theory and why it has been discovered independently in production and utility theory by several authors. These are the important questions. The device used to answer these questions is a multiple characterization of the Cobb-Douglas function in terms of certain properties. Section 3 comments that other functions have been devised in a similar way by seeking functions with specified properties.

## 1. The Contributions of von Thünen and Mill

Von Thünen's important contribution to the marginal productivity theory of factor pricing is well known (see Schumpeter 1954, 466-67). Paul Douglas himself (1948, 2) stated that von Thünen "was thus the real discoverer of marginal productivity." Of course this theory required the concept of the marginal product. It was the marginal productivity theory of wages that led von Thünen to postulate at one stage the Cobb-Douglas function.

The discovery came about in the following way. Von Thünen recognized the importance of marginal analysis in factor pricing. He stated clearly the concept of the marginal product function for labor and capital. The exemplary precision of his conception of marginal products and total products is given by the following quotation from *The Isolated State*:

The work product, p, is the common product of labor and capital. How then is that part which both factors possess in the common product to be measured? We have measured the productivity of capital in the increase that occurs in the work product through the increase of the capital with which a man works. Here labor is a constant but capital is a variable magnitude. If we keep up this process, but conversely consider capital as a constant and the quantity of labor as growing, when in a large operation increased productivity of labor is brought about in the total product through an increase of the laborers by one, we must determine the share contributed by that laborer.

([1863] 1960, 317)

For a production process with more than one input, a marginal product function is in fact a contour of a production function that states the **[End Page 2]** relationship between the total product (the quantity of the output) and the quantities of one input. The appreciation of the marginal products of two factors led him to a (verbal) statement of the notion of a production function in which output is the joint product of two factors, labor and capital, that can be combined in variable proportions (see von Thünen [1863] 1960, 270-71, 317, 321).

As a part of his experiments on...