posted by klynn .
I'm working on some HW for my MBA Managerial Economics class. I've got one problem down to the end, but I'm stuck. Any help would be greatly appreciated. And, of course, if you could check my work on a & b, that would be excellent. Thanks! :)
1.) The mathematics of consumer equilibrium can be demonstrated using the following logic:
A consumer will spend or save his/her entire income, so that:
I = P1X1 + P2X2 + .......... PnXn
Where I = consumer income
X1X2,..............Xn = quantities of n commodities, one of which is saving
P1P2,.............Pn = respective prices of the n commodities (the implied price of saving would be 1)
The consumer wants to maximize his/her total utility, TU, where:
TU = f(X1,X2,..........Xn)
subject to the above income constraint.
a.) Set up the Lagrangian expression that is to be maximized.
I think (?) the answer to this is:
Ltu = TU + lamba(I - P1X1 - P2X2......PnXn)
b.) Show the partial derivative equations whose solution gives optimal values for X1,X2,.........Xn. Note that the partial derivative of TU with respect to X1 is designated as MUx1.
According to my math, these are the partial derivatives:
dLtu/x1 = dTU/dX1 - lambdaP1
This one would be the marginal utility of x1, I think?
dLtu/xn = dTU/dXn - lambdaPn
And, this one shoul dbe the marginal utility of Xn.
c) Show that consumer equilibrium requires the following relationship:
MUx1/P1 = MUx2/P2 = ....... = MUn/Pn
Interpret the economic meaning of the above relationship.
On this one, I'm completely stuck. I've used the Lagrangian Multipler Technique dozens of times, but it's always had actual figures in it. Without actual numbers, I don't even know where to start. :(
your a) and b) look correct.
For c) you are almost there.
You have Ln = dTUn/dXn - LambdaPn
The first term to the right of = is MUn. Also, at a maxima, the whole term is zero. So you have MUn - LambdaPn=0.
Thus, MUn = LambdaPn. This is true for all goods. So, MUx1=LambdaPx1
Thus MUn/MUx1 = LambdaPn/LambdaPx1
The Lambdas cancel. So MUn/MUx1=Pn/Px1
which can be re-written as MUn/Pn = MUx1/Px1
Ok, I think I get it. Thank you so much!