On the Importance of Being Unimportant: An Analysis of the Paradox in Marshall's Third Rule of Derived Demand

Abstract

Of the four rules about the elasticity of derived demand, stated first by Alfred Marshall (1890, pp. 384-386, 852-853), then clarified by A. C. Pigou (1948, pp. 682-685), only the third rule, concerning the importance of being unimportant has been criticized and modified to any extent. Marshall and Pigou deduced that the greater the proportion of total cost accounted for by a factor of production, the more elastic is the derived demand for that factor. Later J. R. Hicks in The Theory of Wages (1932, pp. 242-245) demonstrated mathematically that Marshall's first, second and fourth rules are correct but that the third rule holds only if the elasticity of commodity demand exceeds the elasticity of substitution between that factor and the other factor of production, or all others taken together. Since Hicks' demonstration it has been generally accepted, until quite recently, that Marshall's third rule does not always hold and that the Hicks caveat or paradox must apply.2 Despite general acceptance of the Hicks paradox, in the many papers on the subject there has been considerable confusion and controversy concerning the reasons behind the exceptions. As Bronfenbrenner (1971, pp. 149-150) noted in the most recent attempt at clarification, there have been several attempts to breathe life into Hicks' exception to Marshall's third rule, but none has been successful. Hicks (1932, p. 246) attempted unsuccessfully a brief explanation of his mathematical results. D. H. Robertson (1958, p. 31) in a later comment was quite vague and probably incorrect in his literary exposition of the problem. Bronfenbrenner (1961) compared the three standard formulae for the elasticity of derived demand. He tested for the validity of Marshall's rules using