[求Consider the following scheme devised by the parent of an overweight

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<P><FONT size=2 face="Times New Roman">Consider the following scheme devised by the parent of an overweight child to reduce the child's consumption of candy bars. The parent offers to sell candy bars to the child at their market price P plus a parental tax t. To induce the child to cooperate, the parent offers to increase the child's allowance by S dollars. If the child accepts the deal, he/she agrees not to buy candy from anyone else.</FONT></P>
<P><?xml:namespace prefix = o /><o:p><FONT size=2 face="Times New Roman"></FONT></o:p></P>
<P><o:p></o:p><FONT face="Times New Roman"><FONT size=2>Will the amount of the tax the parent collects from the child be larger or smaller than the increase in the child's allowance? Explain.</FONT></FONT></P>
<P><FONT size=2 face="Times New Roman"></FONT></P>
<P><FONT size=2 face="Times New Roman">想了很久没有想出来，向大家请教一下：）</FONT></P>

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关键词：OVERWEIGHT following Weight follow parent following increase dollars parent agrees

Suppose the child has income M, then the child will promise to buy candy from the parents if and only if

(M+S) / (P+t)>M / P (1)

==> S > t*M / P (2)

The total tax (T) collected from the child is

T = t* (M+S) / (P+t) ==> t = T* (P+t) / (M+S) (3)

Sub (3) into (2), and use the condition (1), it is easy to show that S > T

The result is, the child will consume more candy, which is partly financed by the parents.

thanks, u r so kind:)

However, why don't u consider all the other goods the child cosuming?

Because (M+S)/P>(M+S)/(P+t), the child will spend all the allowrance on the market instead of buying candy bars from the parent. Am I right?

[此贴子已经被作者于2006-10-22 20:33:24编辑过]

这题目是不是有问题啊?没有给出其他商品以及效用方面的信息,单单就靠简单的t和s来确定?

我认为二楼的解法只是起到了不让其在其他地方购买candy bar的作用,但不能使其减少购买量,而且事实上是增加了购买量.

如果不考虑这道题目的实际意义(减少candy bar的消费),二楼的做法也有待商榷,不能得到如上所说的 'it is easy to show that S > T'.(我认为应该稍稍改动一下,即(M+S) / (P+t)>=M / P ,两者相等的情况也是允许的.)请楼上好好计算一下.最后的结果是T<=(PM+PS)*S/(PM+TM)>=S. S与T 无法比较.

不好意思，前面的解答只考虑了Candy一种商品，所以结论很奇怪。现在加入另外一种商品，比如Apple。如图所示，黑色的BL是加税之前的，红色的是加税并获得最少补贴情况下的BL，也就是临界情况下的BL。显然A和B是无差的，这时候，只要Parents稍稍增加S，就可以使得ChildUtility增加。很明显，A在红线以外。过A做另外一条直线，如图中的蓝线，蓝线过A点意味着Parents把收来的税全部用于补贴Child。那么，Parents只要控制Child的BL在红线和蓝线之间，就可以达到“使Child减少消费Candy”，并且补贴小于Child上缴的税。

这个例子实际上是体现了税收中 Sin Tax 的作用。也同时说明有时候，“家长式”作风是可以使福利改善的（也就是我们常说的Optimal Paternalism）。

另外，有一个前提条件：Child不是极度偏好Candy。

[此贴子已经被作者于2006-10-23 3:54:50编辑过]

thank u!