III.--The Elasticity of Substitution and the Relative Share of a Factor

Abstract

WE owe to Dr. Hicks the extremely interesting proposition that an increase in the supply of any factor will increase its relative share (i.e. its proportion of the National Dividend) if its ' elasticity of substitution ' is greater than unity,' and will diminish its relative share if the elasticity of substitution is less than unity. For much the most satisfactory way of proving the validity of this rule, Dr. Hicks refers the reader to his mathematical Appendix. Here is to be found2 a symbolic definition of elasticity of substitution which gives the appearance of having been dictated by the requirements of algebra rather than of economics, and Dr. Hicks fails to tell us what it really means, while in the text we have to be content with the admission that the case where the elasticity of substitution is unity can only be defined in words by saying that in this case (initially, before any consequential changes in the supply of other factors takes place) the increase in one factor will raise the marginal product of all other factors taken together in the same proportion as the total product is raised.' The purpose of this note is to point out that Dr. Hicks' elasticity of substitution has a very simple meaning and to show how his proposition can be deduced by a very simple line of thought.' For, by a curious coincidence, Dr. Hicks' elasticity of substitution is exactly the same as Mrs. Robinson's elasticity of substitution.' Though Dr. Hicks was apparently unaware that his new conception was capable of so simple an interpretation, it is in fact the proportionate change in the ratio of the amounts of the factors divided by the proportionate change in the ratio of their marginal physical productivities.' Let us now imagine a curve relating the ratio of the amounts of the factors (measured along the x axis) to the ratio of the marginal productivities (measured along the y axis).' Then it follows from the above definition that the elasticity of substitution is equal to the elasticity of this curve. By a well-known proposition the area xy subtended by a point on a curve increases as x increases if the curve has an elasticity greater than unity, and