The Ordinal Utility Error

[Jonathan Barzilai]

The Ordinal Utility Error

Since the operations of addition and multiplication are not applicable on ordinal scales, ordinal utility functions cannot be differentiated — the derivatives (ordinal “marginal utilities”) do not exist. Conversely, since ordinal scales cannot be differentiated, differentiable scales are not ordinal. To emphasize, utility scales that satisfy differential conditions are not ordinal: when “marginal utilities” exist, the utility scales in question cannot be ordinal.

It follows that the notion that modern economic theory can be founded on ordinal utility theory is a fundamental error. For example, the equilibrium conditions of demand theory are stated in terms of “marginal utilities” which are partial derivatives of utility functions. If the utility scales of demand theory are ordinal they cannot be differentiated; if they are differentiable they are not ordinal.

The claim that ordinal temperature scales are sufficient to carry out partial differentiation in thermodynamics is obviously false. For the same reasons that the mathematical theory of thermodynamics cannot be founded on ordinal temperature scales, modern economic theory cannot be founded on ordinal data.

The claim that modern economic theory can be founded on ordinal utility theory appears throughout the literature of modern economic theory. See for example Mas-Colell, Whinston, and Green’s Microeconomic Theory. The source of this error is Pareto’s statement (in sections 5—6 of the Appendix to his 1905 Manual of Political Economy) that “The entire theory of economic equilibrium is independent of the notions of (economic) utility.” This error was amplified by Hicks (1930’s & 1946) and Samuelson (1948). Hicks, saying that “Pure economics has a remarkable way of producing rabbits out of a hat,” proceeded “to undertake a purge, rejecting all concepts which are tainted by quantitative utility” and Samuelson’s analysis in his Foundations of Economic Analysis is also based on an incorrect mathematical argument. For details see Ordinal Utility and Indifference Curves, pp. 1—7, 2009, at http://ScientificMetrics.com/publications.html

Over more than a century, utility theorists, economists, measurement theorists, psychologists, and mathematicians have not detected this error.

Jonathan Barzilai