Lagrange Multipliers

[Pauline]

Dear everyone,

I have a problem which I am not fully able to solve. Hence i need your help.

Here’s the question

“ Each month, a given consumer is buying CDs and books. This consumer has a utility functionu(x, y) = x4y5, where x is the number of CDs bought per month, and y is the number of booksbought per month. The price of a CD is 10 dollars, while the price of a book is 15 dollars.This person’s monthly budget for such entertainment purposes is 882 dollars. How many booksand CDs does he buy monthly, if he wants to maximize his utility ? (You may assume that thecritical point obtained corresponds to a relative maximum.) "

I came up with the following answer :

u(x,y) = x⁴ y⁵
x = # of CD
y = # of books

P_xX + P_yY = Income

10 x + 15y = 882

u (x,y) = 10x + 15y -882 = 0
F(x,y,lamda) = x⁴y⁵ - lamda (10x+15y-882)

Partial derivatives
f(x) 4x³ y⁵ - lamda (10)=0
f(y) x⁴ 5y⁴ - lamda (15)=0
f(lamda) -10x-15y+882 =0

Here’s where I am stuck how can i define the least cost input combination ?

Thank you for your answer.

Pauline N.

 

[Pauline]

First why are you trying to define the least cost input combination? The problem asks for the combination that can be afforded with $882, that maximizes the utility function.

What you've done so far is basically correct (other than some poor notation for partial derivatives).

You just have to solve this system of equations and determine the maxima.

It is solvable by hand. Solve the 3rd equation first to get y in terms of x and then plug that in to equations 1 and 2 and solve. The answers won't be pretty but you will be able to evaluate them.

Note that the solution to the system isn't the answer to the problem because you can't buy a fraction of a book or cd. So you have to find the whole number solution that maximizes u(x,y). You'll have to plug a couple of points into your utility function to see which point maximizes it.

What you might do...

a) recognize that you can't buy a fraction of a book
b) plot your constraint curve, i.e. y=(882-x/10)/15
c) plot your utility function on the same graph (scaling it drastically so you can see both, I had to divide it by 1 trillion)

d) eyeball where the maximum is and start plugging in values to find exactly where it is, remember they have to be whole numbers.

That should get you a solution pretty quickly.

Thank you for your answer,

I am trying to solve it with y in terms of x, but I am really confused since there’s lamda.
Could you please compute it for me. I would understand better, at least to solve equation 1 and 2.

I will be able to evaluate it afterwards myself.

Thank you in advance.

Pauline

 

[Pauline]

forget about \(\displaystyle \lambda\) a sec.

you have a constraint equation \(\displaystyle 10x+15y\leq 882\)

solve that for y in terms of x.

It's a line going from (0, 882/15) to (882/10, 0). I.e. going from all books, to all cds. (do you get this?)

You are trying to maximize your utility function. It's an increasing function in x and y so clearly you want to buy as many books and/or cds as you can with your $882. That means your solution is going to be on top of your allowable region, i.e. somewhere on that line, or as close as possible given that you can't buy fractional books and cds.

You can do the same thing with your utility function. Forget about \(\displaystyle \lambda\) a sec and just plot it. Like I said you'll have to scale it by a factor of a trillion or so to put it on the same plot as your constraint equation.

You can eyeball that plot to get a good idea of what x is at the peak of your utility function.

Your set of partial derivatives (now you can worry about \(\displaystyle \lambda\)) gives you the equations for the exact solution, but again you can't buy fractional books and cds. So you just have to get close and try a couple points of integer values and see which combo actually maximizes your utility function, making sure that you actually can afford that combo, i.e. that it's on or below your constraint line.

The reason this isn't a straightforward lagrange multiplier problem is that you can't buy fractional books and cds. If it was something you could buy a continuous amount of, like say salt and pepper, then you'd just solve your set of equations for {x, y, \(\displaystyle \lambda\)} and be done with it, but you can't quite do that here.

See if you can make further headway now.

[addendum]
It occurs to me I might upset your teacher by telling you to eyeball and try solutions, even if it's easier than solving the systems of equations. So here's how you'd solve that system. Or at least some hints.

From the partial with respect to \(\displaystyle \lambda\)

\(\displaystyle \begin{align*}&0=-882+10x+15y \\ &882-10x=15y \\ &y=\dfrac{882-10x}{15}\end{align*}\)

From the partial with respect to y.

\(\displaystyle \begin{align*}&0=5x^4 y^4+15\lambda \\ \\ &0=5x^4 \left(\dfrac{882-10x}{15}\right)^4+15\lambda \\ \\ &15\lambda=5x^4 \left(\dfrac{882-10x}{15}\right)^4 \\ &\lambda=\dfrac{5x^4 }{15}\left(\dfrac{882-10x}{15}\right)^4\end{align*}\)

Now you have \(\displaystyle y\) and \(\displaystyle \lambda\) in terms of \(\displaystyle x\) you can plug that whole mess into the equation from the partial with respect to \(\displaystyle x\) and solve it for \(\displaystyle x\) and then use that solution to find \(\displaystyle y\) and \(\displaystyle \lambda\). But again this will only be approximate as you can't buy fractional amounts, and you'll be right back at trying a few close solutions.

Thank you so much !
I got it this time. Wasn't sure with the computation of "lamda" and "y" to finally find the "whole mess"
but it's easy now to plug it in the main equation.

I understood thanks again!
Pauline N.