Kuhn-Tucker conditions; intermediate microeconomics

[btmagnetw]

can someone explain what the following kuhn tucker conditions mean? this is for consumer utility problems, p=price of good, U(x)=utility, and x=amount of goods. lambda is the lagrangian multiplier, which apparently represents the marginal utility of income, but i can't figure out how to use it.

[tabako]

I'm surprised you are doing this in intermediate micro... where do you go to school?

It should be clear that p and x* should both be vectors, and p.x* is that dot product of them. For (2): you didn't define what y is, but I assume its the consumer's budget or something like that. p.x* is a scalar, and it's how much the consumer spends on goods, so having y - p.x* >= 0 fits with y being the budget.

From this, we can look at the fourth statement and see that one of two things are 0.

either lambda = 0, or y - p.x* = 0.

This says that either the consumer exhausts all of his budget (i.e. y = p.x*), or the marginal utility of income is 0 (i.e. lambda = 0). This should make intuitive sense, as marginal utility of income can't be positive if you aren't using all of your income.

As for the other two statements, I don't have a good grasp of what they intuitively represent, though I might be able to figure out how to solve a problem with them.


I haven't actually done this stuff before, but I am fairly confident that what I said about (2) and (4) is correct.

[tabako]

You might want to message JaredL or econophile, they are (were?) econ grad students and should be able to give you a much better explanation.

[btmagnetw]

thanks for the help. i took this class at UT austin before, and this never came up. i'm studying abroad in Hong Kong now, and i'm taking it again since my friend is, and i figured i could breeze through it the second time. mostly i am, but the professor has included a lot of stuff like this that isn't even in the book.

ok, so if i read the first statement as:

the marginal utility of the highest bundle minus lambda* times the price is less than or equal to 0.

is that correct? i don't understand why it should be, but i just want to see if i'm reading it right.

[tabako]

Earlier this semester, I was enrolled in the first graduate micro class. I didn't have the math background to continue with the class, so I dropped it after a couple weeks. I did see stuff like this in the text, and am surprised you are doing it in intermediate micro. Do you have any sample problems that are to be solved using these conditions?

[tabako]

What do you mean by highest bundle?

I read the first statement as:
For every good Xi in the vector bundle X, the marginal utility of good i is less than lambda* times the price of that good. (Normally I find these things make more sense when instead of A - B < 0, you change it to A < B)

I have no idea why this would hold, though.

[RR]

[ QUOTE ]
Earlier this semester, I was enrolled in the first graduate micro class. I didn't have the math background to continue with the class, so I dropped it after a couple weeks. I did see stuff like this in the text, and am surprised you are doing it in intermediate micro. Do you have any sample problems that are to be solved using these conditions?

[/ QUOTE ]

There are a lot of people that believe this stuff should be in the intermediate classes (I agree). A lot of departments have removed the math from the undergraduate classes to attract more students. As far as the OP it has been over 10 years since I did this so I just don't remember. If I feel really bored today I will look it up.