求助：证明位似偏好的一些性质

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求助：那位高手帮忙证明一下位似偏好的一些性质的证明：

Homothetic preferences can be represented by a utility function that is homogeneous of degree one. That is, U(ax)=aU(x) for all x in the consumption set and all a>0, such that "ax" is in the consumption set. Assuming an interior solution for the consumer’s UMP, show that for this representation of homothetic perferences:

the uncompensated system of demend functions, x(p.w)and the indirect utility function, v(p,w) are homogenous of degree one in w. That is, x(p,w)=wx(p,1), and v(p,w)=wv(p,1).

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关键词：Presentation consumption Compensated homogeneous represented 性质 偏好

(1)Suppose first thar u(.) is homogeneous of degree one and let , ,and .then u(x)=u(y) and hence au(x)=au(y). by the homogeneity, u(ax)=u(ay). Thus .

(2)suppose conversely that is homothetic. Let and a>0, then u(x)e x, and u(ax)e ax. since is homothetic, au(x)e ax. By the transitivety of , u(ax)e au(x)e. thus u(ax)=au(x)

(1)Suppose first thar u(.) is homogeneous of degree one and let a>=0,x属于R ,and x和y无差异.then u(x)=u(y) and hence au(x)=au(y). by the homogeneity, u(ax)=u(ay). Thus ax和ay无差异。.

(2)suppose conversely that 偏好 is homothetic. Let ,x属于R and a>0, then u(x)e与x无差异, and u(ax)e与ax无差异. since 偏好 is homothetic, au(x)e与ax无差异. By the transitivety of无差异, u(ax)e和au(x)e无差异. thus u(ax)=au(x)