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问一个

Minimize wL+rK

subject to: y=f(L,K),

where y=f(L,K)is the production function, L=labor, K=capital, w=wage, and r=price of capital. Answer the following questions for the particular case where f(L,K)=LK/100 and w=r=$1 

a)Use the substitution technique to find the minimizing quantities of labor and capital L* and K*

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关键词：substitution particular Quantities production Technique 题目

use first order condition, don't forget to add constraints. It is called Lagrange function, perhaps.

actually， I don't think Lagrange is necessary to solve this problem.

CAN ANYONE HERE give me your answers? thank you~~

You have constraint, right? So you need to define a lagrange function, and solve its first order condition.

Actually, I think you just want someone else could do it for you. And you just free-ride....something like that....

You need to minimize the cost that is wL+rK, given a certain(or say constant) output (or say production) y=f(L,K) (Surely, the output is determined by the labour and capital input).

So you need to define Lagrange.

I suggest you should do it yourself. You can paste your answer here, so that someone else could check it for you. So you will know how to do it correctly, and why you are wrong if it happens.

The most important thing is that you know, but not that someone else knows.

i'm not sure whether this y is given.cause if it is known, we could use it to transform the extremum with a constraint into an extremum without a constraint instead of using Lagrange
my answer is K=L=10√y~~

I didn't get what you said... Obviously, y could be regarded as given variable. But y is unknown. The question is to minimize the cost in the sense that y, w and r are given, L and K can be freely chosen.

First you should derive the minimizer L(y,w,r) and K(y,w,r). They are functions of y, w and r.

Then insert the f(L,K)=LK/100 and w=r=$1 into it and see the result.

That's all!

if y is given ,then Lagrange may be not necesary.

can you judge my answer is right or wrong?

OK....I give you my answer:

Define Lagrange function:

l = wL + rK - lambda * (f(L,K)-y)    PS. I will supress L and K in f

the minimizer L* and K* must satisfy the FOC:

dl/dL = w - lambda* df/dL =0   ==> w = lambda* df/dL

dl/dK = r - lambda* df/dK =0   ==> r = lambda* df/dK

dl/dlambda = f - y =0   ==> f = y

Now we have the general form of L* and K*, let f = LK/100. We have

df/dL = K/100  (1)

df/dK =L/100   (2)

substitution gives:

K* = 10*sqrt(wy/r)

L* = 10*sqrt(ry/w)

You get the right answer, but I think your lecturer prefers to see all the details.