Monotone Mappings

Abstract

In a seminar given in 1934 Abraham Wald was the first to use the weak axiom of revealed preference (1936a, b). However, in several ways his use of it looks odd to the post-Samuelson reader. First, since Wald’s main purpose was to establish a new condition for unique solution of the modified Walras–Cassel system of general equilibrium he had introduced earlier (1935), as originally stated it was a restriction on market rather than individual behaviour. Secondly, the axiom referred not to the (vector) market demand function z = f(p) but to its inverse p = f⁻¹(z), whose existence is of course quite suspect. Finally, although later in the paper Wald did in fact invoke the individual version (wa) of the weak axiom as some ground – ‘a statistical probability’ – for belief in its market version (WA), he did not justify (wa) as did Samuelson (1938), as in its own right a sensible rule for consistent market behaviour. Instead, Wald derived if from an assumed additive Jevonian utility function for the individual, i.e. u_i(z_i) = Σ _ju_ij(z_ij) where in addition ${\mathrm{d}}^2{u}_{ij}\left({z}_{ij}\right)/\mathrm{d}{z}_j^2<0$ for each person i and good j; and so in Wald (wa) appeared as much more restrictive than it really is.