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.

[btmagnetw]

by highest, i meant optimal bundle. x represents a bundle of goods x1 and x2. so graphing x1 on the x axis and x2 on the y axis, we have indifference curves that represent where the consumer is at the same utility. then we have a budget line, and the optimal bundle is the point where an IC is tangent to that budget line.

i don't have an example problem, since this stuff isn't in the book, and the only material i have of them are the lecture notes. but from class, it seems that a problem could be "show that the optimal bundle of some given utility function and some range of income and prices satisfies the kuhn tucker conditions."

so i'm guessing that he'll give something like U(x) = x1x2 and give values for y, p1, and p2 and i have to plug them into the conditions?

[JaredL]

It's been a couple years since I've done this but hopefully I can provide some info.

Firstly, we're doing constrained optimization. So you are finding the best bundle you can subject to not spending more than your income (y in your setup).

The easiest place to start is (3). The first equation says that for x_i one of two things must be true. Either you purchase none of it or you buy just enough so that the marginal benefit equals the marginal cost. The marginal benefit is the partial derivative of U wrt x_i. The marginal cost is how much of your budget you lose which is p_i (the marginal monetary cost of buying good x_i) times the marginal utility of income. Think of p_i as the amount of money you lose and lambda as the price of that loss.

The reason for this restriction is that the price of good x_i may be such that you would optimally purchase none of the good. Think about how the price of Ferraris affects your purchasing decisions. Similarly, think about how many more ferraris you will buy if you get $1 richer. So in the case of a Ferrari your marginal benefit is probably very high but the price is so high that equation 1 will be strictly negative. Because you won't purchase any, equation 3 is satisfied.

Now look at the second equation in (3). Lambda, which as you said is marginal benefit of income, times the difference between how much money you have and how much you are spending must be zero. This is simply saying that if more money is useful to you (lambda > 0) then you are spending all of your money (y - p*x = 0). Most utility functions are increasing so all money will be spent.

So the punchline here is that you know in the optimal bundle either a positive amount of each good will be consumed, in which case you can solve by setting the marginal utility of x1 divided its price equal to the same for x2 and then plugging that in to p1x1 + p2x2 = y. Or it could be at a corner. The only way to know in general is to check. Some utility functions feature infinite marginal utility for each good at 0 so that the corners are never optimal, but otherwise you just have to check them to make sure that you don't get more utility from a corner than the interior optimum.

[HajiShirazu]

Wow, I had no idea you could use the lagrangian formalism for this stuff. As weird/impractical as it is, it beats the usual "a particle constrained on the surface of a sphere of radius a is in the potential csc^2(bz)" garbage I'm used to.