anonymous
  • anonymous
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Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
anonymous
  • anonymous
@ganeshie8
ganeshie8
  • ganeshie8
Never did these before but I think the budget constraint must be \[60G+6M \le 450\]

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anonymous
  • anonymous
Yeah. Im unsure about the parts B and C though
Astrophysics
  • 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
  • ganeshie8
We have inequalities as constraints and it must be solved over integers right ?
Astrophysics
  • Astrophysics
Yeah that sounds good I think
ganeshie8
  • ganeshie8
The regular lagrange multipliers method wont work here
Astrophysics
  • Astrophysics
What if we just say 60G + 6M = 450 as the budget constraint
Astrophysics
  • Astrophysics
Then we can set up our langranian as \[L(G,M, \lambda) = G^{1/2}+M^{1/2}+\lambda(450-60G-6M)\]
ganeshie8
  • 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
  • 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
  • ganeshie8
http://mat.gsia.cmu.edu/classes/QUANT/NOTES/chap4/node6.html
ganeshie8
  • 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
  • Astrophysics
You're right I made an assumption haha.
ganeshie8
  • 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
  • Astrophysics
Yes, that's right. So it seems this requires an extra step then from the link you provided, checking the complementarity conditions
ganeshie8
  • ganeshie8
yeah we need to include constraints for nonnegativity too
ganeshie8
  • ganeshie8
Maximize \(U(G,M)=G^{1/2}+M^{1/2}\) subject to : \(60G+6M \le 450\) \(-G \le 0\) \(-M\le 0\)
Astrophysics
  • 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
  • Astrophysics
But you still have the optimality conditions
ganeshie8
  • ganeshie8
idk, never worked langrange multipliers with inequality constraint
Astrophysics
  • Astrophysics
Yeah I'm just sort of reading the link you sent, never seen this before either
Astrophysics
  • Astrophysics
https://www.youtube.com/watch?v=3VQBVf6Tr3Y https://www.youtube.com/watch?v=uuXSTsrFo-k

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