Re: [請益] Walrasian budget set 的 Convexity 問題

[welly (insight)]

※ 引述《washburn (Just a game)》之銘言：
By definition, If X is a convex set, then
: x'' = alpha*x + (1-alpha)*x' belongs to LR+, for any alpha belongs to [0,1].

To show B{p,w} is convex, for any a, b belong to B{p,w},
we need to show that c= alpha*a + (1-alpha)*b also belongs to B{p,w}.

First, c belongs to LR is obvious.
Second, p*c=p*(alpha*a + (1-alpha)*b)
           =alpha*p*a+(1-alpha)*p*b
           <=alpha*w + (1-alpha)*w
           <= w
By definition, B{p,w} is a convex set.
Q.E.D.

:     Walrasian budget set：B {p,w} = {x belongs to LR: p*x <= w}。
:     請問，為什麼在 X 為 convex 時，B {p,w} 也是 convex？


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※ 編輯: welly           來自: 122.124.125.58       (01/22 00:31)
推 washburn:你好快!