求大神讲解一道微观经济学题目

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我是专业金融学的大一学生，在学微观经济学，遇到一道题目着实不会，论坛里大神多，求大神帮忙讲解一下，学生在此谢过~

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关键词：微观经济学 经济学题目 微观经济 经济学 大一学生 经济学

Cobb-Douglas function.
Using lagrangian function to solve the problem.
when a+b=1, a and b are the share of the income on purchasing X1 and X2.

X1(P1,P2,M)=aM/[(a+b)P1],
X2(P1,P2,M)=bM/[(a+b)P2].
Thus,P1*X1=aM/(a+b),this is the money spent on X1,then the share of budget is P1*X1/M=a/(a+b),thus when a+b=1,a is the share of income spent on X1.

这是柯布道格拉斯效用函数，见图

annozk 发表于 2014-3-21 22:46
Cobb-Douglas function.
Using lagrangian function to solve the problem.
when a+b=1, a and b are the ...

Thanks for your help.

电气生学金融 发表于 2014-3-21 23:49
这是柯布道格拉斯效用函数，见图

大神，看了你给的答案后我又想了想，您是这样做的不（见图）？要不是的话，能不能给个步骤？十分滴感谢你~~~

As a matter of fact, I used the Lagrangian funtion.
L=X1^a*X2^b-λ(P1 X1+P2 X2-M)
The λ is called Lagrangian Multiplier. Take the derivatives w.r.t X1 and X2 and λ respectively, you will have three equations, these are called First-Order Conditions (FOCs).
  FOCs:
L(X_1 )=aX1^(a-1)*X2^b-λP1=0   ❶
L(X_2 )=bX1^a*X_2^(b-1)-λP2=0       ❷
Lλ=-(P1 X1+P2 X2-M)=0    ❸
Solve the system of these three equations, you will get the solutions.
X1(P1, P2, M) =aM/((a+b) P1 )
X2(P1, P2, M) =  bM/((a+b) P2 )
The λ is the marginal utility of income.
This is the standard procedure of utility maximization problem.

annozk 发表于 2014-3-23 05:09
As a matter of fact, I used the Lagrangian funtion.
L=X1^a*X2^b-λ(P1 X1+P2 X2-M)
The λ is calle ...

Oh~~I get it.   It's kind of you to write the detailed processes for me. Thank you very much.

路过遇上 发表于 2014-3-23 11:53
Oh~~I get it.   It's kind of you to write the detailed processes for me. Thank you very much.

U R welcome!

路过遇上 发表于 2014-3-22 16:55
大神，看了你给的答案后我又想了想，您是这样做的不（见图）？要不是的话，能不能给个步骤？十分滴感谢你 ...

不好意思，好久没上了，才看到你的回复，是的，这是科布道格拉斯函数的推导过程，要构建拉格朗日函数：L=clnx1+clnx2-λ（p1x1+p2x2-m）然后微分
dL/dX1=c/x1-λp1=0 dL/dx2=d/x2-λp2=0 dL/dλ=p1x1+p2x2-m=0解得c=λp1x1 d=λp2x2 所以c+d=λ(p1x1+p2x2)=λm可得λ=m/c+d将此式带入c与d解得x1与x2的需求函数结论就是x1=cm/(c+d)p1
x2=dm/(c+d)p2,c和d分别代表购买商品x1与x2所占收入的比例