# Obj function yielding independent goods demand functions

I know that if the objective function (aka utility) is homothetic, demand functions will be linear in income. So for an homothetic demand function to give goods independent of prices other than their own, one has to have a Cobb-Douglas function as it also has to be homogeneous of degree zero.

My question: can someone supply an example of a class of preferences yielding price independent demand functions $x_i = g(\frac{y}{p_i})$? (no prices nor income in preferences, just quantities)

[Challenge to think, not to give long proofs please. Just sketch proofs with the important steps if feel it's important]

• If I have understood your question correctly you might look at demand systems characterized by price independent generalized linearity (introduced by Muellbauer) such as PIGLOG – DornerA Jun 20 '16 at 20:21
• Using the primal consumer's problem (no prices nor income in preferences, just quantities). Will update question. – dugo Jun 22 '16 at 11:40

So if we take a simple example with 2 goods and consider the consumer's problem:

$L=u(x,y)+\lambda(I-p_xx-p_yy)$

This yields the ratio of FOC: $u_1/u_2=p_x/p_y$

Now plugging these guys into the budget:

$I=p_xu_1^{-1}(u_2p_x/p_y)+p_yy$

A sufficient condition is $u_1^{-1}$ be hod -1 to make $p_x$ drop out. But I don't think it's necessary.

Now I'm not exactly sure what that means (the inverse of the marginal utility of a good is homogeneous of degree -1), but a great thing to think about.