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!

For part c), we need to show that consumer equilibrium requires the relationship: MUx1/P1 = MUx2/P2 = ....... = MUn/Pn. This relationship is known as the marginal rate of substitution (MRS).

To understand the economic meaning of this relationship, let's start by interpreting each term:

- MUx1: Marginal utility of good X1 is the additional utility gained from consuming one additional unit of good X1.
- P1: Price of good X1 represents the amount of money needed to purchase one unit of good X1.
- MUx2: Marginal utility of good X2 is the additional utility gained from consuming one additional unit of good X2.
- P2: Price of good X2 represents the amount of money needed to purchase one unit of good X2.
- and so on for other goods (Xn)...

In consumer equilibrium, the consumer wants to maximize their total utility (TU) subject to the income constraint. The consumer allocates their income (I) across various goods (X1, X2, ..., Xn) to achieve the highest level of satisfaction.

When the consumer is in an equilibrium state, the consumer has optimized their allocation of goods such that the marginal utility per dollar spent is equal for all goods. This means that the consumer is getting the same amount of additional utility for each dollar spent on any of the goods.

Mathematically, this equilibrium condition is represented by the relationship: MUx1/P1 = MUx2/P2 = ....... = MUn/Pn.

To interpret the economic meaning of this relationship, let's focus on two goods, X1 and X2, and their respective marginal utility and prices:

MUx1/P1 represents the additional utility gained per dollar spent on good X1. If MUx1/P1 is higher than MUx2/P2, it implies that the consumer can get more satisfaction per dollar spent by consuming more of good X1 and less of good X2. Conversely, if MUx1/P1 is lower than MUx2/P2, the consumer can increase their overall satisfaction by consuming less of good X1 and more of good X2. When MUx1/P1 = MUx2/P2, the consumer has optimally allocated their resources to maximize their total utility within the given prices and income constraint.

In simpler terms, the consumer equilibrium condition states that the consumer should allocate their income in such a way that the satisfaction derived from the last dollar spent on each good is the same. This ensures that the consumer is maximizing their total utility given the prices of the goods.