Bounding the Price Space ℝ ⁿ₊ by a Collar-Preserving Homeomorphism

Summary

Homogeneity is a frequently used assumption in economics. However, in some cases it is desirable to relax it, particularly if exogenous variables are not proportionally changed. But then the boundedness property of the price domain, which is rather convenient for mathematical treatment, gets lost. In order to recapture the boundedness it is reasonable to map the new price domain homeomorphically onto (a subspace of) the old one. Clearly, such a homeomorphism should be intuitive from the geometrical viewpoint. From the economic viewpoint in particular it should transform the boundary properties of the economic behavior functions “equivalently”. In geometrical terms the last requirement means that the homeomorphism should be “collar-preserving”. The paper provides an intuitive geometrical representation of a collar-preserving homeomorphism together with its inverse. The geometric representation will also be put into exact analytical terms in a way which makes heuristics explicit.