Question

image attached

Answer 1

@ganeshie8

Answer 2

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

Answer 3

Yeah. Im unsure about the parts B and C though

Answer 4

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

Answer 5

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

Answer 6

Yeah that sounds good I think

Answer 7

The regular lagrange multipliers method wont work here

Answer 8

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

Answer 9

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

Answer 10

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 ?

Answer 11

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.

Answer 12

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

Answer 13

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

Answer 14

You're right I made an assumption haha.

Answer 15

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.

Answer 16

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

Answer 17

yeah we need to include constraints for nonnegativity too

Answer 18

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

Answer 19

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

Answer 20

But you still have the optimality conditions

Answer 21

idk, never worked langrange multipliers with inequality constraint

Answer 22

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

Answer 23

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

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