Econ: Given demand $x_1$, $x_2$ under locally non-satiated preferences find the demand for $x_3$

Question

I have a sort of theoretical question that has an application to a problem I am doing. I have been asked the question in the title. Given a three good economy on $\mathbb{R}^3$ and a consumer's Marshallian demand for goods $x_1$, $x_2$ I need to find the demand for $x_3$ given the fact that the consumer is maximizing locally non-satiated preferences.

So consider two demand functions $x_1(p, y)$ , $x_2(p,y)$ that I am given. I want to find $x_3(p,y)$. I believe that under local non-satiation we know the solution lies on the budget constraint ie

$$x \cdot p = y$$

or

$$x_1(p, y)p_1 + x_2(p,y)p_2 + x_3(p,y)p_3 = y$$

Which means we can solve for $x_3(p,y)$ trivially.

$$x_3{p,y} = \frac{1}{p_3} \left( y - x_1(p, y)p_1 + x_2(p,y)p_2\right)$$

Is the correct approach?

Answer 1

You are correct, and the property that you used is called "Walras' law", refering to the 'Preposition 3.D.2' in MWG (1995).

Formally, if the underlying preferences are locally non-satiated, then for all $x \in x(p, y)$, the solution to the UMP (Utility maximization problem), we have $$ p \cdot x = y $$

It can be easily proved as below.

Suppose not, that is, for some $x \in x(p, y)$, $p \cdot x < y$, then there must exist another consumption bundle $x'$ which is sufficiently close to the original bundle $x$ with both $p \cdot x' < y$ and $ x' \succ x$. However, this contradicts the fact that $x$ is the solution to the UMP.