中级微观经济学的问题，跟效用有关的，大家帮帮忙吧。

Question

David has to work in order to earn a living. He is paid an hourly wage.He use his income to purchase various neccessities of life. For the sake of simplicity, suppose that David’s consumption needs are fulfilled by one “composite” good called”c“ . He has to divide his time between work and leisure, but he dislikes work and enjoys leisure. He can devote at most 24 hours a day to leisure. Therefore, if he wants to enjoy leisure for L hours,he can work for only 24-L hours. Suppose that David’s preferences for consumption and leisure are given by the utility function U=(C,L)such that he derives positive marginal utility from both commodities. Also suppose that the price of ”c“ is $1 per unit and the wage rate is $w per hour. Further suppose that David’s wage rate of $ w per hour is for the first eight hours a day and he receives an overtime wage of $w‘ per hour for extra time he works, such that w’＞w . The relevant budget constraints are shown in the figure.
a. If David’s preferences are represented by an indifference curve like U1 , would he choose to work for more than eight hours? Explain your answer.b. If, instead, David’s preferences are represented by an indifference curve like U2, would he choose to work over time? Explain your answer?

Answer 1

大卫工作是为了谋生。他是支付一小时的wage.He用他的收入来购买生活的各个neccessities。为了简单起见，假设大卫的消费需求是一个“复合”好称为“C”履行。他要分他的工作和闲暇时间，但他不喜欢的工作，享受休闲。他可以花24小时，最多一天休闲。因此，如果他想享受休闲的L小时，他能只有24 - L的时间。假设消费和休闲大卫的偏好是由效用函数u =（C和L），使得他来自两种商品的边际效用给予积极的。还假设的“C”的价格是每单位的工资率是每小时$ 1瓦特。进一步假设，大卫的工资率每小时$ W是前8小时，一天，他收到了$加班工资w'的每小时额外的时间，他的作品，使得W“>瓦特。相关的预算约束是如图所示。
答：如果大卫的偏好是由一个像U1的无差异曲线表示，他会选择工作八个多小时？解释你的answer.b。相反，如果大卫的偏好是由一个像U2的无差异曲线表示，他会选择在工作一段时间？解释你的答案吗？

Answer 2

直接从图上可以直观地看出结果：
a.utility和直线的交点，横坐标大于16，说明david prefers working less than 8 hours. So he wouldn't choose work for more than 8 hours.
b.utility和直线的交点，横坐标小于16，说明david prefers working more than 8 hours. So he would choose to work over time.

Answer 3

图上的16表示闲暇时间，也即工作8小时的时间点。然后由于超时工作后会加工资，所以实际的工资预算线应为w‘ 闲暇为0-16时
w 闲暇为16-24时
是一条折线。
当效用用无差异曲线U1表示的时候，最佳效用最大化点在16-24之间，即U1和w的交点处，其他点的效用都比该点低或者说无法达到。所以他不会超时工作
当效用用U2表示时，首先，在工资预算线（之前所说的折线）外的点都是无法达到的点，因此，对于U2，只有U2与w’相交两点之间的点才是可以选择的点，它们显然在0-16之间。这意味着他会选择超时工作