x1的反需求函数是:p1?mp22m?ifx1? x12p2x1的恩格尔曲线:x1?是参数.
右上图中红色线(x2?2m其中,m是自变量,p1,p22p1?p2x1x2 12mx1,x1?)是价格提供曲线 2p2右下图中绿线是收入提供曲线.x2?1x1 2x1 X1是normal good, ordinary good, and supplementary good for x2. (3)
bmax{x1ax2}st..
(求最大化的过程同第8题,这里从略) x1的需求函数:x1?amam(其中p1,p2,m为自变量), 反需求函数:p1?,
(a?b)p1(a?b)x1恩格尔曲线:x1?
am(其中m为自变量)
(a?b)p1右图中,红线为p1价格提供曲线,(x2?bm)
(a?b)p2兰线为收入提供曲线(注意,这里收入提供曲线是直线)
x1是normal good, ordinary good, 和x2没有总替代或互补关系. (4)
max{lnx1?x2}s....t
最大化求解过程同第8题,这里略去.
X1的需求函数:当m?p2时,x1的需求函数是:x1?p2m; 当m?p2时,x1的需求函数是:x1? p1p1p2m; 当m?p2时,x1的反需求函数是:p1? x1x1 X1的反需求函数:当m?p2x1的反需求函数是:p1? 11 / 33
?p2?p?1恩格尔曲线:x1???m??p1ifm?p2
ifm?p2右图中,红线为m>1时的p1价格提供曲线(x2=m-1);
绿线为m<1时的p1价格提供曲线( x2=0)(假设p2=1) 蓝线为收入提供曲线
x1是normal good,ordinary good. 是x2的总替代品。.
10. In this problem, we focus on the Slutsky substitution effect only.
Suppose the utility function isu(x1,x2)?x1?ax2,a?0.initially the prices of thecommodities arep1and p2, respectively, and the wealth of the consumer, m.
First, assume
?m?p11?, so that the initial consumption bundle is?,0?. Then the prices vary.Withoutloss of generality, p2a?p1?assume the price of commodity 1 varies fromp1top1?.
Case 1.
?m?p1?1
?, so that the final consumption bundle is?,0?.
?p??p2a?1?
Since under the final prices, given that the initial bundle is just affordable, the consumer picks exactlythe initial bundle as well, so that the own price substitution effect for commodity 1 is zero. And theincome effect ispositive if the price of commodity 1 becomes less, vice versa.
Case 2.
mm?, which is p1?p1p11?, so that any bundle satisfyingp1?x1?p2x2?mis probably selected.Suppose that finally the bundle?x1,x2?is p2a 12 / 33
chosen by the consumer.Under the final prices, and given the initial bundle can be just affordable, there are also infinitebundles which may be selected. Assume now, the consumer picksx1?,x2?. Then the substitutioneffect for
????commodity 1 isx1Case 3.
m? , and the income effect is x1?x1p1?m?p11?, so that the final bundle chosen by the consumer is?0,?.Since, under the new prices, the initial bundle is p2a?p2??mp??1?. So, the substitution effect for commodity 1 also exactly affordable, the bundle picked by theconsumer is ?0,?p1p2???is?m, and the incomeeffect for commodity 1 is zero. p1Now, assume initially
?m?p11?, so that the initial bundle is?0,?, then the price ofcommodity 1 becomes p1?. p2a?p2?Analogously, the followings hold:
Case 4.
?m?p1?1mIf , and the income effect is ?, the final bundle is?,0?. And the substitution effect for commodity 1 is
??p2ap1??p1??
zero.
Case 5.
p1?1If ?, and assume the final bundle selected by the consumer is?x1,x2?, thensubstitution effect for commodity 1
p2a
isx1, and the income effect is zero. Case 6.
p1?1If ?, the final bundle is then the initial bundle. Then both the substitution effectand income effect for commodity
p2a
1 are zero.
And then suppose initially
p11?, then any bundle satisfying p1x1?p2x2?mis probablyselected, assume p2athat?x1,x2?is initially chosen. Case 7.
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?m?p?x?p2x2pxp1?1
?x1?22, If ?, then the final bundle is?,0?.The substitution effect for commodity 1 is 11?p??p2ap1?p1??1?
and the income effect is
mp2x2. ??p1p1?p1?1
And note, it’s impossible that ?. Otherwise the price of commodity 1 doesn’t vary at all.
p2a
Case 8.If
?m?p1?1
?, then the final bundle is?0,?. Under the final prices, if the initial bundle is just affordable, the p2a?p2??p1x1?p2x2?consumer shall select thebundle?0,?, so the substitution effect for commodity 1 is ?x1, and the
p?2?incomeeffect is zero. (End)
11．It is easy to find the initial consumption of y is
m30m30??7.5, the final one is ??10, and the bundle 2py43?2pychosen if one faces the budget line with the same slope of the final budget line and through the initial consumption bundle, which is
pxx?p?yy2py?26.25?8.75. 3So 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 2fixed, 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 WARPis 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
Then R*?8973, L*?, c*?89 99 14 / 33
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
Then R*?29, L*?25, c*33?116 (End)
maxu?c1c214．(1)
s.. tcc2m2 1?1?r?m1?1?rL?cm21c2??(m1?1?r?cc1?21?r) f.o.c. c2??,c1??1?r
c?1?1454.5,c?2?1600,s??m1?c?1?2000?1454.5?545.5,
(2) Similarly, we can get
c?1?1416.7,c?2?1700
maxu?c1c215．(1) maxs.. tc21?c21?r?mm
1?1?rL?c21c2??(m1?m1?r?cc21?1?r) f.o.c. c?2??,c1?1?r
c?1?600,c?2?450
maxu?c1c2(2)
s.. tcc2m2
1?1?r?m1?1?rL?cm21c2??(m1?1?r?cc1?21?r) f.o.c. c2??,c1??1?r
c??1?568.2,c2?625.
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