Cycle-Preserving Extension of Demand Functions to New Commodities

A method is given to extend demand functions to new commodities under preservation of the cycle number, i.e., the minimal length of a preference cycle revealed by the demand function. Thus Gale's (Economica, N.S., 1960, 27, 348-354) demand function that shows that the weak axiom of revealed preference does not imply the strong axiom of revealed preference for three commodities can be extended to more than three commodities. Also Shafer's (Journal of Economic Theory, 1977, 16, 293-309) result, that arbitrarily high cycle numbers exist for three commodities, can now be extended to any number of commodities larger than three. This completely settles a question raised by Samuelson (Economica, N.S., 1953, 20, 1-9).