1) In the town of Springfield, residents work at home and receive $100,000 of income
1） 在斯普林菲尔德镇，居民在家工作并获得 100,000 美元的收入

each year unless their house burns down, in which case they receive $0. Half of the
每年，除非他们的房子被烧毁，在这种情况下，他们收到 0 美元。一半的

residents are descended from the Flanders clan, and are by nature careful; their
居民是佛兰德斯氏族的后裔，生性细心;他们

probability of fire is 0.2% each year. Half of the residents are descended from the
火灾的概率为每年0.2%。一半的居民是

Simpson clan, and are not as cautious. Their probability of fire is 1.0% per year.
辛普森家族，并没有那么谨慎。他们的火灾概率是每年 1.0%。

The residents also have different utility functions. The descendents of Flanders have
居民也有不同的公用事业功能。佛兰德斯的后裔有

utility:

UF=(X+5)^¼

Where X is their income. The descendents of Simpson are from two lines—half
其中 X 是他们的收入。辛普森的后代来自两个血统——一半

descended from Bart and the other half from Maggie. Bart Simpson’s descendents have
巴特的后裔，另一半是玛姬的后裔。巴特·辛普森（Bart Simpson）的后代有

utility:

UBS=(X+5)^½

While Maggie’s have utility:
虽然 Maggie's 有实用性：

UMS=X

Write down each type’s expected utility in a world with no insurance.
写下每种类型在没有保险的世界中的预期效用。

b) The town’s sole insurer, appropriately named Burns Insurance Co., cannot distinguish between the residents. If it offers actuarially-fair insurance based on the town’s average rate of fires, what will be the equilibrium price? Who will buy insurance?
b） 该镇唯一的保险公司，恰当地命名为 Burns Insurance Co.，无法区分居民。如果它根据该镇的平均火灾率提供精算公平的保险，那么均衡价格是多少？谁会购买保险？

c) A new safety device is invented that can cut the risk of fires in half. It costs $5 per
c） 发明了一种新的安全装置，可以将火灾风险降低一半。每个收费 5 美元

year. Assume that Burns has no ability to monitor device use. Assume also that those
年。假设 Burns 无法监控设备使用情况。还假设那些

who are indifferent between buying and not buying the new device don’t buy it. Under these assumptions, who will buy the device? What will be the new equilibrium price of insurance? Who will buy insurance?
那些在购买和不购买新设备之间无动于衷的人不要购买它。在这些假设下，谁会购买该设备？保险的新均衡价格将是什么？谁会购买保险？

d) Compute the change in welfare for each group and society as a whole. Who is better off? Who is worse off? Why?
d） 计算每个群体和整个社会的福利变化。谁过得更好？谁的情况更糟？为什么？

e) Explain how moral hazard and adverse selection affect the equilibrium choice of
e） 解释道德风险和逆向选择如何影响均衡选择

insurance and safety device use in (c).
（c）中的保险和安全装置使用。

f) Now assume Burns can require the use of the device among all the insured. What will be the new equilibrium price of insurance? Who will buy insurance? What is the level of social welfare?
f） 现在假设 Burns 可以要求所有被保险人使用该设备。保险的新均衡价格将是什么？谁会购买保险？社会福利水平如何？

g) Explain the changes in part (f) from part (c).
g） 解释（f）部分与（c）部分的变化。

2) An economy produces two goods - ice cream and hot chocolate. Each of these goods is produced by a separate person. Individual 1 produces ice cream and receives a profit of $64 if it is hot, $0 if it is cold. Individual 2 produces hot chocolate and receives a profit of $0 if it is hot, $64 if it is cold.
2）一个经济体生产两种商品——冰淇淋和热巧克力。这些商品中的每一种都是由单独的人生产的。个人 1 生产冰淇淋，如果是热的，则获得 64 美元的利润，如果是冷的，则获得 0 美元的利润。个人 2 生产热巧克力，如果热巧克力，利润为 0 美元，如果热巧克力，利润为 64 美元。

Individual 1 and 2 maximize expected utility:
单个 1 和 2 最大化预期效用：

E[U1]= p* U(Y1H)+(1-p)* U(Y1C)

E[U2]= p* U(Y1H)+(1-p)* U(Y1C)

where YH and YC are income in state H (hot) and state C (cold), U(Y) is the utility
其中 YH 和 YC 是状态 H（热）和状态 C（冷）的收入，U（Y） 是效用

function, and p is the probability of state H. There is a social welfare function of the
函数，p 是状态 H 的概率。有社会福利功能

form:

Total Welfare=E[U1]+E[U2]

(a) Suppose that p=1/2, and that the form of the utility function is:
（a） 假设 p=1/2，效用函数的形式为：

U(X)=X^1/2

What is the initial expected utility of each individual?
每个人的初始预期效用是什么？

(ii) Suppose that the two individuals are considering entering into an arrangement today which insures their income against this uncertain weather outcome tomorrow (there is no weather forecasting in this society). That is, depending on the weather outcome (state H or C), each person will either receive some of the other person's income, or give some of their income to the other person. Can such an insurance arrangement be struck that makes both parties better off? What arrangement will maximize social welfare? How does social welfare compare to the level in (i)? Is this arrangement acceptable to both parties?
（ii） 假设两个人今天正在考虑达成一项安排，以确保他们的收入免受明天这种不确定的天气结果的影响（这个社会没有天气预报）。也就是说，根据天气结果（状态H或C），每个人都将获得对方的部分收入，或者将部分收入交给对方。能否达成这样的保险安排，使双方都变得更好？什么样的安排才能使社会福利最大化？与第（i）项相比，社会福利水平如何？这种安排对双方都能接受吗？

(iii) Now, suppose a perfectly accurate weather prediction system is invented, and it is
（iii） 现在，假设发明了一个完全准确的天气预报系统，它是

announced that it will be hot tomorrow. How much insurance will be bought and sold
宣布明天会很热。买卖多少保险

now? What expected utility does this give each person? Has this increased or decreased social welfare, relative to (ii)? Why?
现在？这给每个人带来了什么预期的效用？相对于（ii）项，这是否增加了或减少了社会福利？为什么？

(b) Now, suppose that the weather prediction system turns out not to work after all, so
（b） 现在，假设天气预报系统最终无法工作，那么

p=1/2 again. But now assume that the form of U is U(X) = ½ X. Answer (i)-(iii) above. Is your answer to (iii) different from (a)? Why or why not?
p=1/2。但现在假设 U 的形式是 U（X） = 1/2 X.上面的答案 （i）-（iii）。你对（iii）的回答与（a）不同吗？为什么或者为什么不？