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anonymous

  • one year ago

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  1. anonymous
    • one year ago
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  2. anonymous
    • one year ago
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    @ganeshie8

  3. ganeshie8
    • one year ago
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    Never did these before but I think the budget constraint must be \[60G+6M \le 450\]

  4. anonymous
    • one year ago
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    Yeah. Im unsure about the parts B and C though

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

  6. ganeshie8
    • one year ago
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    We have inequalities as constraints and it must be solved over integers right ?

  7. Astrophysics
    • one year ago
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    Yeah that sounds good I think

  8. ganeshie8
    • one year ago
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    The regular lagrange multipliers method wont work here

  9. Astrophysics
    • one year ago
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    What if we just say 60G + 6M = 450 as the budget constraint

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

  11. ganeshie8
    • one year ago
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    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 ?

  12. Astrophysics
    • one year ago
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    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.

  13. ganeshie8
    • one year ago
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    http://mat.gsia.cmu.edu/classes/QUANT/NOTES/chap4/node6.html

  14. ganeshie8
    • one year ago
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    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

  15. Astrophysics
    • one year ago
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    You're right I made an assumption haha.

  16. ganeshie8
    • one year ago
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    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.

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

  18. ganeshie8
    • one year ago
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    yeah we need to include constraints for nonnegativity too

  19. ganeshie8
    • one year ago
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    Maximize \(U(G,M)=G^{1/2}+M^{1/2}\) subject to : \(60G+6M \le 450\) \(-G \le 0\) \(-M\le 0\)

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

  21. Astrophysics
    • one year ago
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    But you still have the optimality conditions

  22. ganeshie8
    • one year ago
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    idk, never worked langrange multipliers with inequality constraint

  23. Astrophysics
    • one year ago
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    Yeah I'm just sort of reading the link you sent, never seen this before either

  24. Astrophysics
    • one year ago
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    https://www.youtube.com/watch?v=3VQBVf6Tr3Y https://www.youtube.com/watch?v=uuXSTsrFo-k

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