<p>I don't know what is the definition of the &quot;law of DMRS&quot;, but I think what the author meant is summarized in these lines of MWG:</p>
<blockquote>
<p>Proposition 3.G.2: suppose that <span class="math-container">$u(\cdot)$</span> is a continuous utility function representing a locally nonsatiated and strictly convex preference relation <span class="math-container">$\succsim$</span> defined on <span class="math-container">$\mathbb{R}^L_+$</span>. Suppose also that <span class="math-container">$h(\cdot, u)$</span> [the hicksian demand function] is continuously differentiable at <span class="math-container">$(p, u)$</span>, and denote its <span class="math-container">$L\times L$</span> derivative matrix [with respect to each <span class="math-container">$p$</span>] by <span class="math-container">$D_p h(p, u)$</span>. Then</p>
<ol>
<li><p><span class="math-container">$D_p h(p, u)$</span> is a negative semidefinite matrix.</p>
</li>
<li><p><span class="math-container">$D_p h(p, u) p = 0$</span></p>
</li>
</ol>
</blockquote>
<p>Well, 1 implies that the elements of the main diagonal of the matrix are all non positive, i.e. <span class="math-container">$\frac{\partial h_l (p, u)}{\partial p_l} \leq 0$</span> for each <span class="math-container">$l$</span>.</p>
<p>We say that two goods are substitutes if <span class="math-container">$\frac{\partial h_l (p, u)}{\partial p_k} \geq 0$</span> (a positive variation in the price of <span class="math-container">$k$</span> implies an increase in the compensated demand of <span class="math-container">$l$</span>).</p>
<p>Part 2 of the proposition says that <span class="math-container">$D_p h(p, u) p = 0$</span>. But as <span class="math-container">$\frac{\partial h_l (p, u)}{\partial p_l} \leq 0$</span>, for every <span class="math-container">$l$</span> we must have a good <span class="math-container">$m$</span> such that <span class="math-container">$\frac{\partial h_l (p, u)}{\partial p_m} \geq 0$</span>, which means exactly that every good has a substitute.</p>

CC BY-SA 4.0
2019-04-14T19:06:48.300Z
2019-04-14T19:06:48.300Z
2020-06-17T09:40:54.643Z
Marcelo Gelati
<p>I've been reading a book where the author claims that "if there are two goods they are always net substitutes. Along an indifference curve an increase in price of good 2 leads to less of good 2 and more of good 1 being consumed when utility is maximized. When there are more than two goods, the law of diminishing marginal rate of substitution implies that at least one good must be a net substitute for any other good." 
<br/>
The first part can be proved easily using graphs, it's the second that I couldn't prove mathematically.
<br/>
Source of the quotation has been attached <a href="https://books.google.co.in/books?id=pi4JBAAAQBAJ&amp;pg=PA104&amp;lpg=PA104&amp;dq=net%20substitutes%20more%20than%20two%20goods%20law%20of%20diminishing%20marginal%20rate%20of%20substitution%20implies%20that%20at%20least%20one%20of%20the%20goods%20must%20be%20a%20net%20substitute%20for%20any%20other%20good&amp;source=bl&amp;ots=rvPf6etHY5&amp;sig=AoDLRTE3tZYYM5r9KaFgKFIpx5Y&amp;hl=en&amp;sa=X&amp;ved=2ahUKEwjFzpWgzpndAhXbe30KHbbRBlUQ6AEwAHoECAQQAQ#v=onepage&amp;q=net%20substitutes%20more%20than%20two%20goods%20law%20of%20diminishing%20marginal%20rate%20of%20substitution%20implies%20that%20at%20least%20one%20of%20the%20goods%20must%20be%20a%20net%20substitute%20for%20any%20other%20good&amp;f=false" rel="nofollow noreferrer">here</a>, Pg-104.</p>

<microeconomics><self-study>
Prove that in presence of many goods, the law of DMRS implies that at least one good must be a net substitute for any other good
Chd
# Prove that in presence of many goods, the law of DMRS implies that at least one good must be a net substitute for any other good

(Asked by: username_0 on 2018-09-01)

I've been reading a book where the author claims that "if there are two goods they are always net substitutes. Along an indifference curve an increase in price of good 2 leads to less of good 2 and more of good 1 being consumed when utility is maximized. When there are more than two goods, the law of diminishing marginal rate of substitution implies that at least one good must be a net substitute for any other good."
  
The first part can be proved easily using graphs, it's the second that I couldn't prove mathematically.
  
Source of the quotation has been attached [here](https://books.google.co.in/books?id=pi4JBAAAQBAJ&pg=PA104&lpg=PA104&dq=net%20substitutes%20more%20than%20two%20goods%20law%20of%20diminishing%20marginal%20rate%20of%20substitution%20implies%20that%20at%20least%20one%20of%20the%20goods%20must%20be%20a%20net%20substitute%20for%20any%20other%20good&source=bl&ots=rvPf6etHY5&sig=AoDLRTE3tZYYM5r9KaFgKFIpx5Y&hl=en&sa=X&ved=2ahUKEwjFzpWgzpndAhXbe30KHbbRBlUQ6AEwAHoECAQQAQ#v=onepage&q=net%20substitutes%20more%20than%20two%20goods%20law%20of%20diminishing%20marginal%20rate%20of%20substitution%20implies%20that%20at%20least%20one%20of%20the%20goods%20must%20be%20a%20net%20substitute%20for%20any%20other%20good&f=false), Pg-104.

**Tags:** microeconomics, self-study

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## 1 answers

### Answer by username_1 on 2019-04-14. Score: 2

I don't know what is the definition of the "law of DMRS", but I think what the author meant is summarized in these lines of MWG:

> Proposition 3.G.2: suppose that $u(\cdot)$ is a continuous utility function representing a locally nonsatiated and strictly convex preference relation $\succsim$ defined on $\mathbb{R}^L\_+$. Suppose also that $h(\cdot, u)$ [the hicksian demand function] is continuously differentiable at $(p, u)$, and denote its $L\times L$ derivative matrix [with respect to each $p$] by $D\_p h(p, u)$. Then
>
> 1. $D\_p h(p, u)$ is a negative semidefinite matrix.
> 2. $D\_p h(p, u) p = 0$

Well, 1 implies that the elements of the main diagonal of the matrix are all non positive, i.e. $\frac{\partial h\_l (p, u)}{\partial p\_l} \leq 0$ for each $l$.

We say that two goods are substitutes if $\frac{\partial h\_l (p, u)}{\partial p\_k} \geq 0$ (a positive variation in the price of $k$ implies an increase in the compensated demand of $l$).

Part 2 of the proposition says that $D\_p h(p, u) p = 0$. But as $\frac{\partial h\_l (p, u)}{\partial p\_l} \leq 0$, for every $l$ we must have a good $m$ such that $\frac{\partial h\_l (p, u)}{\partial p\_m} \geq 0$, which means exactly that every good has a substitute.


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