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- anonymous

##### 1 Attachment

- anonymous

- ganeshie8

Never did these before but I think the budget constraint must be
\[60G+6M \le 450\]

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## More answers

- anonymous

Yeah. Im unsure about the parts B and C though

- Astrophysics

\[\nabla f(x,y) = \lambda \nabla g(x,y)~~~~~\text{and}~~~~~g(x,y) = k\] this is the lagrange multiplier, where g(x,y) = k would be the constraint.

- ganeshie8

We have inequalities as constraints and it must be solved over integers right ?

- Astrophysics

Yeah that sounds good I think

- ganeshie8

The regular lagrange multipliers method wont work here

- Astrophysics

What if we just say 60G + 6M = 450 as the budget constraint

- Astrophysics

Then we can set up our langranian as \[L(G,M, \lambda) = G^{1/2}+M^{1/2}+\lambda(450-60G-6M)\]

- ganeshie8

then we can use the plain old lagrange multipliers but the problem doesn't say he spends full 450, so don't you think we're changing the problem by simplifying it ?

- Astrophysics

Well it says endowment of 450 dollars that he spends on buying games and digital music, it's sort of sounds like he is spending all of it...but you could be right to, not sure.

- ganeshie8

http://mat.gsia.cmu.edu/classes/QUANT/NOTES/chap4/node6.html

- ganeshie8

your budget for a week is $450 doesn't necessarily mean you will be spending all of it, it just means that you cannot exceed $450. if this problem is from equality constraints then ofcourse equality constraint makes sense. otherwise it doesn't

- Astrophysics

You're right I made an assumption haha.

- ganeshie8

I mean, the optimal utility need not happen on the surface of g(x,y) = k, it "can" happen anywhere inside the solid g(x,y) <= k.

- Astrophysics

Yes, that's right. So it seems this requires an extra step then from the link you provided, checking the complementarity conditions

- ganeshie8

yeah we need to include constraints for nonnegativity too

- ganeshie8

Maximize \(U(G,M)=G^{1/2}+M^{1/2}\)
subject to :
\(60G+6M \le 450\)
\(-G \le 0\)
\(-M\le 0\)

- Astrophysics

So it's kind of like solving for two constraints \[L(G,M, \lambda_1, \lambda_2) = G^{1/2}+M^{1/2}+\lambda_1(-G)+\lambda_2(-M) \]

- Astrophysics

But you still have the optimality conditions

- ganeshie8

idk, never worked langrange multipliers with inequality constraint

- Astrophysics

Yeah I'm just sort of reading the link you sent, never seen this before either

- Astrophysics

https://www.youtube.com/watch?v=3VQBVf6Tr3Y
https://www.youtube.com/watch?v=uuXSTsrFo-k

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