1??1p2??3?0 ② p1x1?p2x2?m,x1?0,x2?0 ③ ?1?0,?3?0,?2?0

互补松弛条件：?1(m?p1x1?p2x2)=0 ④ ?2x1=0 ⑤

?3x2=0 ⑥ 1??3由②知：?1?>0 ，所以由④知：p1x1?p2x2?m ⑦

p2m Ⅰ。如果?3>0，则x2=0，所以由⑦有 x1?>0，从而?2=0

p11?1? 再由①有 ?1???

2?mp1? 由② ?3??1p2?1??3?12p2?1????1 2?mp1?12122p2?1?p2 ?3必须满足?3>0，所以，? ??1>0?m<

2?mp1?4p12mp2 所以当m<时，x1?，x2?0

p14p1Ⅱ。?3=0，则x2>0，由①知x1?0，所以?2=0，由因为?3=0，所以由②知2p1mp2?1??2，因为x2>0，所以，代入①得，x1?2，x2?p2p24p14p12p2mp2?>0?m> p24p14p122pmp2p2?2。 所以，当m>时，解为：x1?2，x2?p24p14p14p1mp1x1? 大家也可以通过预算约束把x2表示成x2?，然后代入到效用函数中讨论p2p2其极值。

max{2x1?x2}9. (1)

s.tp1x1?p2x2?m?mifp1?2p2?p?1?m?ifp1?2p2 商品一的需求函数为:x1??[0?]p1??0ifp1?2p2???

2 右图中,红色线为价格提供曲线.

x1的收入提供曲线,当p1?2p2时,是横轴

x当p1?2p2时,是整个第一像限 当p1?2p2时,是纵轴

m?mifx?1?x2p2?1m?ifx1?(0,] 反需求函数是:p1??2p22p2??[2p2,??)ifx1?0??0 x1 恩格尔曲线:如果p1?2p2那么恩格尔曲线是:x1?m p1m],?m 2p2 如果p1?2p2那么恩格尔曲线是一个柱面: x1?(0, 如果p1?2p2那么恩格尔曲线是:x1?0,?m

x1是正常品(normal,相对于劣等品而言), 是一般商品(ordinary,相对于Giffen品而言)

x2是替代品(其实是完全替代品)

max{min(x1,2x2)}(2)

st.p1x1?p2x2?m2mx2x1需求函数:x1?其中p1,p2,m是自变量 2p1?p2 mp2mx1的反需求函数是:p1??2ifx1?

x12p2x1 x1的恩格尔曲线: x1?变量,p1,p2是参数. 右上图中红色线(x2?供曲线

2m其中,m是自

2p1?p2x2 12mx1,x1?)是价格提2p21x1 2X1是normal good, ordinary good, and

x1 supplementary good for x2.

bmax{x1ax2}(3)

st..(求最大化的过程同第8题,这里从略)

amamx1的需求函数:x1?(其中p1,p2,m为自变量), 反需求函数: p1?,

(a?b)p1(a?b)x1am恩格尔曲线: x1?(其中m为自变量) (a?b)p1右下图中绿线是收入提供曲线. x2?

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. max{lnx1?x2}(4)

s....t最大化求解过程同第8题,这里略去.

pX1的需求函数:当m?p2时,x1的需求函数是:x1?2; 当m?p2时,x1的需求函数

p1m是: x1?

p1p X1的反需求函数: 当m?p2x1的反需求函数是:p1?2; 当m?p2时,x1的反需

x1m求函数是: p1?

x1?p2ifm?p2?p?1恩格尔曲线:x1??

?mifm?p2??p1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 the commodities are p1 and p2 , respectively, and the wealth of the consumer, m.

First,

?m?p11?, so that the initial consumption bundle is?,0?. Then p2a?p1?the prices vary. Without loss of generality, assume the price of assume

commodity 1 varies fromp1to p1?.

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 exactly the initial bundle as well, so that the own price substitution effect for commodity 1 is zero. And the income

effect is

mm?, which is positive if the price of commodity 1 becomes ?p1p1less, vice versa.

Case 2. p11?, so that any bundle satisfying p1?x1?p2x2?m is probably selected. p2aSuppose that finally the bundle?x1,x2?is chosen by the consumer. Under the final prices, and given the initial bundle can be just affordable, there are also infinite bundles which may be selected. Assume now, the consumer picksx1?,x2?. Then the substitution effect for commodity 1

??isx1m? , and the income effect is x1?x1p1??Case 3.

?m?p11?, so that the final bundle chosen by the consumer is?0,?. Since, p2a?p2?under the new prices, the initial bundle is also exactly affordable, the

?mp??bundle picked by the consumer is ?0,1?. So, the substitution effect for

?p1p2???mcommodity 1 is?, and the income effect for commodity 1 is zero.

p1?m?p11?, so that the initial bundle is?0,?, then the p2a?p2?price of commodity 1 becomes p1?. Analogously, the followings hold: Now, assume initially Case 4.

?m?p1?1

?, the final bundle is?,0?. And the substitution effect for If

?p??p2a?1?

mcommodity 1 is, and the income effect is zero.

?p1Case 5.

p1?1

?, and assume the final bundle selected by the consumer is?x1,x2?, If

p2a

then substitution effect for commodity 1 isx1, and the income effect is zero. Case 6.