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If a demand function $X(P_X, P_y, I)$ depends only on prices and income. Is this function a homogeneous function?

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All (marshallian) demand functions are homogenous of degree zero in prices and income. This is a well known result.

That means that $x(\lambda p, \lambda I) = x(p,I)$, where $p$ is a vector of prices (i.e. x could be a function of any number of prices, including $P_x$ and $P_y$, which were given in your example).

So if we multiply all prices and income by the same amount, demand doesn't change. This is very intuitive. Imagine the government would just add a zero to all prices and to your paycheck. There is no reason for you to change your demand as nothing really got cheaper nor did your income increase in real terms.

To see this formally:

Proof:

$x(p,I) = argmax \; U(x)$ s.t. $px \leq I$.

$x( \lambda p, \lambda I) = argmax \; U(x)$ s.t. $ \lambda px \leq \lambda I \Leftrightarrow $

$x( \lambda p, \lambda I) = argmax \; U(x)$ s.t. $px \leq I = x(p,I)$ $Which \; is \; the \; definition \; of \; homogeneity \; of \; degree \; zero. QED. $

The last step follows because we cancel $\lambda$ in the budget constraint on both sides.

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With no other information, it is not possible to determine with $X(P_X, P_y, I)$ is a homogenous function. As mentioned in the answer by @BB King, it is (very) likely that because it is a demand function it is homogenous of degree 0.

Formally, a function, $f : X \rightarrow \mathbb{R}$, is homogenous (of degree $d$) if, for any constant $c$, $f(cx) = c^d f(x)$. Using this definition will allow you to check for yourself whether it is homogenous or not.

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