Market demand independent of distribution of income

Question

If preferences are identical and homogeneous, then show that market demand for any good must be independent of the distribution of income.

My workings are as follows: $$q^{d}(p)=\sum_{i=1}^{n}f(p_x,p_{-x},I)$$

$$q^{d}(p)=f(p_{x},p_{-x})\sum_{i=1}^{n}I$$ Here, $x$ is the price of that particular good and $(-x)$ is the price vector of all the rest of the goods except $x$. Market demand is given by the submission of all consumers from $1$ to $n$.
But, I'm still confused how this function is multiplicatively separable in the price vector $P$.Can someone give a proof of it. Thanks!

Answer 1

I take homogeneity of preferences here to mean that preferences are homothetic; that is, preferences can be represented by a utility function that is homogeneous of degree one.

For an arbitrary good $x$ and a vector of prices $\mathbf{p}$, let individual demand be $x_i(\mathbf{p}, M_i)$ for each consumer $i$ with income $M_i$. Thus market demand is $D = \sum_{i} x_i(\mathbf{p},M_i)$. Since preferences are identical,

$$\sum_{i} x_i(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},M_i).$$

Then, by homogeneity,

$$\sum_{i} x(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},1) M_i = x(\mathbf{p},1) \sum_{i} M_i = x(\mathbf{p},1)M,$$

where $\sum_i M_i = M$ is aggregate consumer income. Thus $D$ is independent of the distribution of income and is solely a function of aggregate income across consumers, $M$.