To address this proof in the context of Walrasian demand functions $x(p, w)$ and Walras' Law, let's go over the statements given and their implications.

Definitions and Notation

1. Walras' Law states that for any price vector $p$ and wealth level $w$, the demand function satisfies $p \cdot x(p, w) = w$.
2. The Walrasian (or Marshallian) demand function $x(p, w)$ describes the amount of each good demanded given prices $p$ and wealth $w$.
3. $D_p x(p, w)$ is the derivative of the demand function with respect to prices, and $D_w x(p, w)$ is the derivative with respect to wealth. These are typically represented as matrices (or Jacobians) when $x$ is a vector-valued function.

Proof of Statements

We need to show:

1. $p \cdot D_p x(p, w) + x(p, w)^T = 0^T$
2. $p \cdot D_w x(p, w) = 1$

Step-by-Step Proof

1. Proving $p \cdot D_p x(p, w) + x(p, w)^T = 0^T$

Using Walras' Law, $p \cdot x(p, w) = w$, differentiate both sides with respect to $p$:

$\frac{\partial}{\partial p} (p \cdot x(p, w)) = \frac{\partial}{\partial p} w = 0$

Applying the product rule to the left side:

$x(p, w) + p \cdot \frac{\partial x(p, w)}{\partial p} = 0$

This can be rewritten in matrix form as:

$p \cdot D_p x(p, w) + x(p, w)^T = 0^T$

This completes the proof for the first part.

Matrix Dimensions for Part 1

• $p$: A vector of prices with dimension $1 \times n$ (if there are $n$ goods).
• $D_p x(p, w)$: The Jacobian matrix of $x(p, w)$ with respect to $p$, with dimension $n \times n$.
• $x(p, w)$: The demand vector, with dimension $n \times 1$.
• Result: The expression $p \cdot D_p x(p, w)$ will yield a $1 \times n$ vector, and $x(p, w)^T$ is also $1 \times n$, so their sum is a $1 \times n$ zero vector.

2. Proving $p \cdot D_w x(p, w) = 1$

Differentiate Walras' Law with respect to $w$:

$\frac{\partial}{\partial w} (p \cdot x(p, w)) = \frac{\partial}{\partial w} w = 1$

Applying the chain rule on the left side:

$p \cdot \frac{\partial x(p, w)}{\partial w} = 1$

or

$p \cdot D_w x(p, w) = 1$

Matrix Dimensions for Part 2

• $p$: A $1 \times n$ vector.
• $D_w x(p, w)$: A $n \times 1$ vector, representing the derivative of demand with respect to wealth.
• Result: The product $p \cdot D_w x(p, w)$ is a scalar, matching the right-hand side (1).

Summary

Both parts are proven, and the matrix dimensions match as expected:

1. $p \cdot D_p x(p, w) + x(p, w)^T = 0^T$ with $1 \times n$ dimension.
2. $p \cdot D_w x(p, w) = 1$ as a scalar.

Would you like further explanation on any part of this proof, or do you have specific questions?

Here are some related questions that might deepen understanding:

1. What is the economic interpretation of Walras' Law in terms of market equilibrium?
2. How does the Slutsky equation relate to this proof?
3. Why is the Jacobian matrix important in demand analysis?
4. What would happen if we had a non-Walrasian demand function?
5. How do these proofs apply to different types of utility functions?

Tip: Always check the dimensions of matrices in economic proofs to ensure that each step aligns with the expected outcome.