Please any ecönomics student shoul help me to solve this.
Proof that
mu1/p1=mu2/p2=mu3/p3

Jenkoja: Please any ecönomics student shoul help me to solve this.
Proof that
mu1/p1=mu2/p2=mu3/p3

Hey @Jenkoja

I just saw this. I hope it is not too late.

Your question concerns what we call Consumer Equilibrium Theory. In that theory, we always seek to maximize the total utility of consuming a particular good/service or a bundle of goods/services as in your case(with 3 goods/services). That is, we seek to find the quantity of goods/services to consume in order to maximize our total utility. An economic agent that thinks this way is said to be rational (although there are other axioms of rationality, I will not disturb you with them).

Now, in maximizing this total utility, the economic agent faces one very important constraint(there are others, please note). This constraint concerns the economic agent's income or wealth. It is called a budget constraint. Essentially, the economic agent cannot consume goods/services more than the income in his possession. In order words, the goods/services must be chosen in such a way that the cost of purchasing them would not be more than the economic agent's overall income. I think you get my point. For instance, if you have 10 naira and you want to purchase goods from a supermarket, you cannot buy goods whose cost is more than the 10 naira you have. This is essentially what is happening.

We now formulate the Optimization problem. But before that, I want you to recall two things:

1. The economic agent wishes to maximize his total utility of consuming a particular good (or 3 goods in your case)
2. The economic agent faces a constraint (budget constraint) in that he cannot consume more than his income

With the above in mind, we formulate the problem as follows:
1. X= bundle of goods=(x1, x2, x3). That is, there are 3 goods
2. P=vector of prices of these goods=(p1, p2, p2)', i.e. pi is the price of one unit of good xi, for i=1,2,3
3. We assume that no two goods have the same price, otherwise there will be a cancellation of prices. So, we assume that p1 is not equal to p2 is not equal to p3
4. Furthermore, we assume that the economic agent's income or wealth is w

So, the economic agent maximizes the utility U(X)=U(x1, x2, x3) for x1, x2, x3 (remember, the aim is to find x1, x2, x3 that maximizes the agent's utility), subject to the constraint that the amount spent on these goods must be at most the agent's income w, i.e. XP=p1x1+p2x2+p3x3<=w or w>=p1x1+p2x2+p3x3. Remember, p1 is the cost of a unit of good x1; so, x1 must cost p1x1. The same argument holds for x2 and x3.

So, the Optimization problem is:

Max U(x1, x2, x3) for x1, x2, x3
Subject to p1x1+p2x2+p3x3<=w or w>=p1x1+p2x2+p3x3 (any one you like. They are the same)

The next thing is to write down the routine Lagrangian:

L= U(x1, x2, x3)+ λ(w-p1x1+p2x2+p3x3)

Take partial derivatives with respect to x1, x2,x3 and set each of these derivatives to zero (see your math-econs courses)

x1: ∂L/∂x1 =∂U(x1, x2, x3)/∂x1 + λ∂(w-p1x1+p2x2+p3x3)/∂x1 = Mu1-λp1=0 ---------> λ=Mu1/p1 (1)
Similarly,
x2: ∂L/∂x2 =∂U(x1, x2, x3)/∂x2 + λ∂(w-p1x1+p2x2+p3x3)/∂x2=Mu2-λp2=0------------>λ=Mu2/p2 (2)
x3: ∂L/∂x3 =∂U(x1, x2, x3)/∂x3 + λ∂(w-p1x1+p2x2+p3x3)/∂x3=Mu3-λp3=0------------>λ=Mu3/p3 (3)

Now, recognize that equations (1), (2), (3) all have the same left hand side, and each of them has equality sign. Thus, it follows that their right hand side must all be equal. Essentially, therefore, we must have

λ=Mu1/p1=Mu2/p2=Mu3/p3
or
Mu1/p1=Mu2/p2=Mu3/p3 as required to prove.


Remark: The derivative of a utility function with respect to quantity of goods/services yields the marginal utility. In elementary economics, we say a change in total utility divided by change in quantity of good(s) equals marginal utility. That is why ∂U(x1, x2, x3)/∂xi=Mui for each i, where i=1,2,3 in your case.

Hope this helps.

T22.