00:01
All right, this is the problem we're given.
00:03
Well, at least i think this is the problem we're given.
00:06
I think there was an issue with the, with uploading your question to numerate.
00:10
I think it should be x squared and y squared.
00:14
It might be the square root, but i think it's x squared and y squared.
00:16
It just ended up looking to me like 250 x2.
00:21
So i'm going to solve it as if it were x squared and y squared because that's actually a sensible question to ask.
00:27
That makes sense to have increasing costs like that.
00:31
Increasing marginal costs.
00:34
So we need to set up our lagrangian.
00:35
We're trying to minimize our costs here.
00:42
So our lagrangian, big l, is going to be this cost part.
00:51
So it's going to be, i can get my pen to work, 250, x squared plus 120, y squared plus lambda, times 1480 minus x minus y so the first part is like our objective that we're trying to minimize or maximize we put there you can do minus lambda here instead of plus lambda it it doesn't really matter it just matters the interpretation of what what exactly lambda is but it's not if you're getting it the same answer either way and we want to set up one side of this equation minus the other side and again that doesn't really matter too much by convention we usually put like the constant first, but you would end up at the same answer either way.
01:48
It would just flip the sign of what this lambda is, the shadow price here.
01:55
Okay, so then we take our first order conditions.
01:58
We have to take a derivative with respect to x of this thing and said it equals zero.
02:01
So that's pretty easy.
02:02
Derivative of this with respect to x is 500x minus lambda.
02:13
That equals zero, right? we have this 250x squared derivative of that is 500x.
02:18
We have a lambda times a minus x.
02:20
So minus lambda.
02:23
The derivative with respect to y is 240 y.
02:31
Whoa.
02:33
I'll do that...