On the Continuity of Utility Functions in the Theory of Revealed Preference
在Uzawa（1971）的基础上，研究显示偏好理论中效用函数的连续性，发现恩格尔曲线的特定性质对效用函数的下半连续性起关键作用，且该性质不直接来自Uzawa的假设。
This paper builds upon Uzawa's [1971] important contribution in the field of revealed preference analysis. One of his remarkable results is that there are conditions implying that the relation indirectly revealed preferred (called R*) is strictly convex and the set {xlx eR'R A xOR*x} for x0 eRn is open in Rn Postulates implying that the set {xlx eR A xR*xO} is open in R too, had been an open problem for a long time, and can be found in articles by Stigum [1973] and Mas-Colell [1978]. Assuming Uzawa's conditions DI-DV in a slightly modified form, it will be shown in this paper that a certain property of the Engel curves associated with the given demand function plays a key role for the lower semicontinuity2 of the utility function generating the given demand function.3 We shall see that this property does not follow from Uzawa's hypotheses alone. If we compare Sonnenschein's Lemma ([1971], p. 77) with our Theorem 1, we shall recognize that condition (c) of his Lemma plays a similar role for the lower semicontinuity of the utility function u generating the given demand function as our restriction on the Engel curves. Both are assumptions on the boundary points of the set {xlx e Rn A u(x') > u(x)} for any x' in the range of the demand function.