p1?1If ?, the final bundle is then the initial bundle. Then both the p2a
substitution effect and income effect for commodity 1 are zero.
p1And then suppose initially 1?, then any bundle satisfying
p2ap1x1?p2x2?m is probably selected, assume that?x1,x2?is initially chosen.
Case 7.
?m?p1?1
If ?, then the final bundle is?,0?.The substitution effect for
?p??p2a?1?
p?x?p2x2pxmpx?x1?22, and the income effect is ?22. commodity 1 is 11?p1p1?p1?p1?p1?1
?. Otherwise the price of commodity And note, it’s impossible that
p2a
1 doesn’t vary at all.
?m?p1?1
, then the final bundle is??0,?. Under the final prices, p2a?p2?if the initial bundle is just affordable, the consumer shall select the Case 8. If
?px?p2x2?bundle?0,11?, so the substitution effect for commodity 1 is ?x1,
p2??and the income effect is zero. (End)
m30??7.5, the 11．It is easy to find the initial consumption of y is 2py4final one is
m2py??30?10, and the bundle chosen if one faces the budget 3line with the same slope of the final budget line and through the initial
pxx?p?26.25yyconsumption bundle, which is ??8.75. 2py3So the substitute effect is 8.75-7.5=1.25, and the income effect is 10-8.75=1.25.(End)
12．For example, the price of commodity 1 decreases with the price of commodity 2 fixed, we can draw a line with the same slope of the new budget line through the initial consumption bundle. And it is easy to find
that any bundles lying on the left side of the new line are less than the initial consumption of x1 and vice versa. If the substitute effects are positive, we will find that the new optimal point lying on the left side, thus the WARP is violated. We can draw a conclusion that WARP supports the negative substitute effect (the law of demand).(End) 13. The problem of the consumer is
maxu?R,c??cR
s..twR?c?wR?m
1) If w=9, R?18,m=16, set up the Lagrange function as
L?cR???m?wR?wR?c??cR???178?9R?c?
f.o.c. R??, c?9?, 9R?c?178
8973Then R*?, L*?, c*?89
99
2) If w??12, then the Lagrange function is
L?cR???m?w?R?w?R?c??cR???232?12R?c? f.o.c. R??, c?12?, 12R?c?232
2925*Then R*?, L*?, c?116
33(End)
maxu?c1c214．(1) c2m2
s.. tc1??m1?1?r1?rmcL?c1c2??(m1?2?c1?2)
1?r1?r1?rc?1454.5,c2?1600,s??m1?c1??2000?1454.5?545.5,
?1?f.o.c. c2??,c1??
(2) Similarly, we can get
c1??1416.7,c2??1700
15．(1) max
c2m2
s.. tc1??m1?1?r1?rmcL?c1c2??(m1?2?c1?2)
1?r1?rmaxu?c1c2 f.o.c. c2??,c1?maxu?c1c2s.. tc1??1?rc1??600,c2??450
(2)
c2m
?m1?21?r1?rmcL?c1c2??(m1?2?c1?2)
1?r1?r1?rc1??568.2,c2??625.
f.o.c. c2??,c1??
16．有一个永久债券（consol）, 每年支付5万，永久支付，利率为r，它在市场出售时价格应是多少？
Solution：First, the interest rate here should be taken as net interest rate. We assume that the 50 thousands yuan is going to be paid at the end of each year from now on。
According to the definition of present value：
?55 Present Value???iri?1?1?r?(End)
17．答：计算储存这瓶红酒在各年的回报率：r1?25?15?0.667，1526?25x?26?0.04，r3??0,x?26。因为利息率i＝5%>r2，所以应在第2525二年初卖掉这瓶红酒。 r2?
18． 课本第四题（review questions）。
40答：PV?B15.42。 10?1.1?19.
?cna?MU?ca??? 已知最优条件，和预算约束1???1???MU?cna??1???ca?2???1????m??L?cna??ca。可得cna???ca，代入预算约束，解出
1??1????1?????m??L??1????2?ca?B28832.06，222??1???????1????m??L??1????cna?22?1????2???1????1???2B34957.38。
20． 答：（1）参加赌博的预期效用是：
EU1?1/2?20000?1/2?0?1002/2，不参加赌博的效用是100，较大。所以，此时不参加赌博。
（2）参加赌博的预期效用是：EU1?1/2?30000?1/2?0?1003/2，
不参加赌博的效用是100，较大。所以仍不参加赌博。
设EU?1/2?10000?x?1/2?0?100?U?10000?，得到x?30000。所以，赢时净挣30000时愿意参加？
21. :Cohb-Douglas效用函数下x，y的需求函数是：
mmx(px,py,m)? y(px,py,m)?
2py2pxx，y价格是1，收入为200时：
200200?100， y(1,1,200)??100， 22消费者的效用 u0?u(100,100)?10,000 x(1,1,200)?x的价格涨至2时：
200200?50， y(2,1,200)??100 42消费者的效用 u1?u(50,100)?5,000 x(2,1,200)?x的价格从1涨至2时，消费者剩余的变化（The lost consumer
2surplus）是：?CS??x(p,1,200)dp??1100dp?100ln2?69.3 p12用C表示补偿变化（Compensating variation）有：
u[x(2,1,m?C),y(2,1,m?C)]?u0200?C200?C??100,000 42?C?200(2?1)?82.8用E表示等价变化（Equivalent variation）有：
u[x(1,1,200?E),y(1,1,200?E)]?u1 or200?E200?E ??5,00022?E?100(2?2)?58.622. Proof：拟线性的效用可以表示成：u(x,y)?v(x)?y
or在预算约束pxx?y?m（把y的价格标准化为1）下，假设内点解，x的反需求函数是：px?v'(x)，由此可见，x的需求与收入无关，在y的价格不变时有：x(px,1,m)?x(px)，
y的需求等于：y?m?pxx(px)
这时消费者的效用水平：u?v[x(px)]?m?pxx(px)
'设x的价格从px变化到px，则消费者剩余变化（The lost consumer
surplus）是：
x(px)'x(px)?CS?[
?0v'(x)dx?pxx(px)]?[?0''v'(x)dx?pxx(px)]'''?{v[x(px)]?m?pxx(px)}?{v[x(px)]?m?pxx(px)}
?u?u'设补偿变化为C有： ''u[x(px,1,m?C),y(px,1,m?C)]?u or'''v[x(px)]?m?C?pxx(px)?u'''C?u?{v[x(px)]?m?pxx(px)?u?u'
?设等价变化为E有：
u[x(px,1,m?E),y(px,1,m?E)]?u' or'v[x(px)]?m?E?pxx(px)?u'
?E?{v[x(px)]?m?pxx(px)}?u'?u?u'
对比可见对于拟线性的效用函数?CS?C?E
第二部分 生产者理论
23.