<p>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. </p>

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

<p><span class="math-container">$$\sum_{i} x_i(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},M_i).$$</span> </p>

<p>Then, by homogeneity, </p>

<p><span class="math-container">$$\sum_{i} x(\mathbf{p},M_i) = \sum_{i} x(\mathbf{p},1) M_i = x(\mathbf{p},1) \sum_{i} M_i = x(\mathbf{p},1)M,$$</span> </p>

<p>where <span class="math-container">$\sum_i M_i = M$</span> is aggregate consumer income. Thus <span class="math-container">$D$</span> is independent of the <em>distribution</em> of income and is solely a function of aggregate income across consumers, <span class="math-container">$M$</span>.</p>

CC BY-SA 4.0
2018-10-12T16:22:11.550Z
2018-10-12T17:09:43.123Z
2018-10-12T17:09:43.123Z
Kenneth Rios
<p>If preferences are identical and homogeneous, then show that market demand for any good must
be independent of the distribution of income.</p>

<p>My workings are as follows: 
<span class="math-container">$$q^{d}(p)=\sum_{i=1}^{n}f(p_x,p_{-x},I)$$</span></p>

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

CC BY-SA 4.0
2018-10-12T17:15:52.757Z
That market demand is price-separable seems to follow immediately from its form; that is, $D = f(\mathbf{p}) \sum_{i}M_i.$ See if my answer helps with your understanding of the derivation.
Kenneth Rios
<microeconomics><supply-and-demand>
Market demand independent of distribution of income
Henam
# Market demand independent of distribution of income

(Asked by: username_0 on 2018-10-12)

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!

**Tags:** microeconomics, supply-and-demand

> **Comments:**
> **username_1**: That market demand is price-separable seems to follow immediately from its form; that is, $D = f(\mathbf{p}) \sum_{i}M_i.$ See if my answer helps with your understanding of the derivation.  

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## 1 answers

### Answer by username_1 on 2018-10-12. Score: 3 | Accepted answer

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, $M$.


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