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# 求救.....經濟學題目,贈20點

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• 8 年前
最佳解答

Ans:Your question is not difficult, but its answer is almost unlikely obtained in this website (Yahoo Knowledge +).

In my experiences, 90% of users in this website are high school students or college students looking for the answer of their homeworks.

How can you expect that they can give you the answer?

To simplify the notations in your question, let α=a, X1=X, X2=Y.

Hence, utility function is U=a*lnX+(1-a)*lnY1. MUx=dU/dX=a/X, MUy=dU/dY=(1-a)/Y,

MRSxy=MUx/MUy = [a/(1-a)](Y/X)2. The first-order condition for equilibrium is MRSxy=Px/Py,

[a/(1-a)](Y/X)=Px/Py ---> Y=[(1-a)/a](Px/Py)X -----(1)

---> Substitute (1) into budget constraint PxX+PyY=I

---> PxX+Py[(1-a)/a](Px/Py)X=I

---> X=aI/Px ----(2) (This is X's demand function)

---> Substitute (2) into (1), we can obtain Y=(1-a)I/Py (Y's demand function) 3. To find out expenditure function, we use the following model:

min PxX+PyY=I s.t. U=a*lnX+(1-a)*lnY=ln[(X^a)(Y^(1-a))] (^表示次方)

--> Again, we use the first-order condition for equilibrium.

---> Substitute (1) into utility constraint U=ln[(X^a)(Y^(1-a))]

---> U=ln{{[[(1-a)/a](Px/Py)]^(1-a)}X}

---> X={[(Py/Px)(a/(1-a))]^(1-a)}exp(U) ---(3)

---> Substitute (3) into (1), we can obtain Y={[(Px/Py)(a/(1-a))]^a}exp(U) ---(4)

---> Substitute (3) and (4) into PxX+PyY, we can obtain the expenditure function:

E=Px*{[(Py/Px)(a/(1-a))]^(1-a)}exp(U)+Py*{[(Py/Px)(a/(1-a))]^(1-a)}exp(U)

---> It can be simplified as E={[(Px/a)^a]*[Py/(1-a)]^(1-a)}exp(U)

.

2012-01-28 00:55:50 補充：

Sorry, the 3rd part shall be revised as follows:

3. To find out expenditure function, .....

min PxX+PyY ( not PxX+PyY=I)

參考資料： myself
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