<p>There is this nice expository note:<br>
Carter M., 1995, "An expository note on the composite commodity theorem," Economic Theory, 5, 175-179<br>
and an interesting generalization by<br>
Lewbel, A., 1996, "Aggregation without Separability: A Generalized Composite Commodity Theorem," American Economic Review, 86, 524-543. </p>

<p><strong>EDIT:</strong><br>
One reason why it is difficult to find a proof of the theorem, is that the so called "composite commodity theorem" does not really have the mathematical status of a theorem. 
It is more a principle directly following from the reparameterization below.<br>
Let <span class="math-container">$x\in \mathbb{R}^J_+$</span> denotes the disaggregate demand system, which depends on the price vector <span class="math-container">$p \in \mathbb{R}^J_+$</span>, and the budget <span class="math-container">$b&gt;0$</span>. The relative prices are denoted by <span class="math-container">$\alpha = p/\textbf{p}$</span> where the aggregate price index is <span class="math-container">$\textbf{p} \in \mathbb{R_+}$</span>. If <span class="math-container">$\alpha $</span> is a vector of constants, we can simplify the demand system such that it depends only upon the aggregate price <span class="math-container">$\bf{p}$</span>: 
<span class="math-container">$$ x(p,b) = x(\alpha \mathbf{p},b) \equiv \mathbf{x}(\mathbf{p},b).  $$</span>
Note that now <span class="math-container">$\mathbf{x}:\mathbb{R^2_+} \rightarrow \mathbb{R}^J_+$</span>.
If one is also interested in aggregating the elementary quantities, it is (for instance) possible to define the aggregate composite quantity as:
<span class="math-container">$$ \mathbf{X}(\mathbf{p},b) = \alpha^T\mathbf{x}(\mathbf{p},b),  $$</span> where <span class="math-container">$ \mathbf{X}:\mathbb{R^2_+} \rightarrow \mathbb{R}_+$</span>. In this case, the elementary and aggregate expenditures are the same, in the sense that: <span class="math-container">$$\mathbf{p}\mathbf{X}(\mathbf{p},b)=p^Tx(p,b).$$</span></p>

CC BY-SA 4.0
2019-06-28T20:55:27.357Z
2019-07-01T11:27:36.663Z
2019-07-01T11:27:36.663Z
Bertrand
<p>I have been reading <em>Economics and Consumer Behavior</em> by Angus Deaton and John Muellbauer, specifically reading up on <strong>Composite Commodity Theorem</strong>, which states:</p>

<blockquote>
  <p>if prices move in parallel to each other, then the corresponding group of commodities can be treated as a single good. (page 120 in the book)</p>
</blockquote>

<p>This is a pretty mind blowing result (well at least for me), however im yet to find an actual proof for the theorem in the literature.</p>

<p>is there a proof for this result?</p>

<microeconomics><mathematical-economics><consumer-theory>
Is there a proof for composite commodity theorem?
EconJohn
# Is there a proof for composite commodity theorem?

(Asked by: username_0 on 2019-06-28)

I have been reading *Economics and Consumer Behavior* by Angus Deaton and John Muellbauer, specifically reading up on **Composite Commodity Theorem**, which states:

> if prices move in parallel to each other, then the corresponding group of commodities can be treated as a single good. (page 120 in the book)

This is a pretty mind blowing result (well at least for me), however im yet to find an actual proof for the theorem in the literature.

is there a proof for this result?

**Tags:** microeconomics, mathematical-economics, consumer-theory


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## 1 answers

### Answer by username_1 on 2019-06-28. Score: 3

There is this nice expository note:  
Carter M., 1995, "An expository note on the composite commodity theorem," Economic Theory, 5, 175-179  
and an interesting generalization by  
Lewbel, A., 1996, "Aggregation without Separability: A Generalized Composite Commodity Theorem," American Economic Review, 86, 524-543.

**EDIT:**  
One reason why it is difficult to find a proof of the theorem, is that the so called "composite commodity theorem" does not really have the mathematical status of a theorem.
It is more a principle directly following from the reparameterization below.  
Let $x\in \mathbb{R}^J\_+$ denotes the disaggregate demand system, which depends on the price vector $p \in \mathbb{R}^J\_+$, and the budget $b>0$. The relative prices are denoted by $\alpha = p/\textbf{p}$ where the aggregate price index is $\textbf{p} \in \mathbb{R\_+}$. If $\alpha $ is a vector of constants, we can simplify the demand system such that it depends only upon the aggregate price $\bf{p}$:
$$ x(p,b) = x(\alpha \mathbf{p},b) \equiv \mathbf{x}(\mathbf{p},b). $$
Note that now $\mathbf{x}:\mathbb{R^2\_+} \rightarrow \mathbb{R}^J\_+$.
If one is also interested in aggregating the elementary quantities, it is (for instance) possible to define the aggregate composite quantity as:
$$ \mathbf{X}(\mathbf{p},b) = \alpha^T\mathbf{x}(\mathbf{p},b), $$ where $ \mathbf{X}:\mathbb{R^2\_+} \rightarrow \mathbb{R}\_+$. In this case, the elementary and aggregate expenditures are the same, in the sense that: $$\mathbf{p}\mathbf{X}(\mathbf{p},b)=p^Tx(p,b).$$


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