<p>MWG proposition 3.G.2 is the one you’re thinking of, but that applies to Hicksian demand’s hessian (constant utility). What you solved for is the Walrasian demand, so we want to look at the Slutsky matrix.</p>
<p><span class="math-container">$$S_{lk}(p,w) = \partial x_l(p,w) / \partial p_k + \partial x_l(p,w)/ \partial w * x_k(p,w)$$</span></p>
<p>First term: uncompensated change in demand of good <em>l</em> due to a change in <span class="math-container">$p_k$</span> (when multiplied by <span class="math-container">$\partial p_k$</span>)</p>
<p>Second term: effect of wealth change on demand of good <em>l</em> (when
multiplied by <span class="math-container">$\partial p_k$</span>) [https://scholar.harvard.edu/files/basilico/files/mwg_flashcards.pdf]</p>
<p>If you check the symmetry of this matrix (in our case <em>l</em> and <em>k</em> are <em>x</em> and <em>y</em>), I believe it should work out.</p>

CC BY-SA 4.0
2019-06-19T15:52:09.113Z
2021-05-25T22:53:28.817Z
2021-05-25T22:53:28.817Z
Kyle Butts
<p>I'm puzzled <em>why</em>, in the following optimal-choice problem, good <span class="math-container">$x$</span> depends on price <span class="math-container">$p_y$</span> but the opposite is not true. Should not the sign of <span class="math-container">$\frac{\partial x}{\partial p_y}$</span> be the same of <span class="math-container">$\frac{\partial y}{\partial p_x}$</span>?</p>

<p><strong>Optimal-choice problem</strong>: given prices <span class="math-container">$p_x$</span> and <span class="math-container">$p_y$</span> and income <span class="math-container">$m$</span> find the optimal demand function for <span class="math-container">$x$</span> and for <span class="math-container">$y$</span>:
<span class="math-container">\begin{equation}
\max_{x,y} \{x\cdot(y+6)\} \qquad \text{s.t. } p_x x+p_y y \leq m \tag{1}
\end{equation}</span>
after imposing <span class="math-container">$MRS_{xy}=p_x/p_y$</span> and substituting back in the budget constraint I find:
<span class="math-container">\begin{equation}
x = \frac{m + 6p_y}{2p_x} \qquad\qquad y = \frac{m}{2p_y}-3 \tag{2}
\end{equation}</span>
so that <span class="math-container">$sign\left(\frac{\partial x}{\partial p_y}\right) = sign\left(  \frac{6}{2p_x}\right)\neq sign\left(\frac{\partial y}{\partial p_x}\right) = 0$</span>.</p>

<p><strong>What I would like to understand better:</strong></p>

<ol>
<li>How can this happen? I remember having read somewhere in MWG that the  price-derivative matrix should be (even!) symmetric. Can you provide me with some explanation and/or some reference? Alternatively, is there any mistake in the calculations?</li>
<li>Provided the situation described above can happen [i.e. <span class="math-container">$sign\left(\frac{\partial x}{\partial p_y}\right)\neq sign\left(\frac{\partial y}{\partial p_x}\right)$</span>], I would be very happy if you can come up with some "real world" examples of that.</li>
</ol>

<microeconomics><consumer-theory>
Asymmetric (in sign) cross-price derivatives in consumer-theory problem
GabMac
# Asymmetric (in sign) cross-price derivatives in consumer-theory problem

(Asked by: username_0 on 2019-06-19)

I'm puzzled *why*, in the following optimal-choice problem, good $x$ depends on price $p\_y$ but the opposite is not true. Should not the sign of $\frac{\partial x}{\partial p\_y}$ be the same of $\frac{\partial y}{\partial p\_x}$?

**Optimal-choice problem**: given prices $p\_x$ and $p\_y$ and income $m$ find the optimal demand function for $x$ and for $y$:
\begin{equation}
\max\_{x,y} \{x\cdot(y+6)\} \qquad \text{s.t. } p\_x x+p\_y y \leq m \tag{1}
\end{equation}
after imposing $MRS\_{xy}=p\_x/p\_y$ and substituting back in the budget constraint I find:
\begin{equation}
x = \frac{m + 6p\_y}{2p\_x} \qquad\qquad y = \frac{m}{2p\_y}-3 \tag{2}
\end{equation}
so that $sign\left(\frac{\partial x}{\partial p\_y}\right) = sign\left( \frac{6}{2p\_x}\right)\neq sign\left(\frac{\partial y}{\partial p\_x}\right) = 0$.

**What I would like to understand better:**

1. How can this happen? I remember having read somewhere in MWG that the price-derivative matrix should be (even!) symmetric. Can you provide me with some explanation and/or some reference? Alternatively, is there any mistake in the calculations?
2. Provided the situation described above can happen [i.e. $sign\left(\frac{\partial x}{\partial p\_y}\right)\neq sign\left(\frac{\partial y}{\partial p\_x}\right)$], I would be very happy if you can come up with some "real world" examples of that.

**Tags:** microeconomics, consumer-theory

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## 1 answers

### Answer by username_1 on 2019-06-19. Score: 3 | Accepted answer

MWG proposition 3.G.2 is the one you’re thinking of, but that applies to Hicksian demand’s hessian (constant utility). What you solved for is the Walrasian demand, so we want to look at the Slutsky matrix.

$$S\_{lk}(p,w) = \partial x\_l(p,w) / \partial p\_k + \partial x\_l(p,w)/ \partial w \* x\_k(p,w)$$

First term: uncompensated change in demand of good *l* due to a change in $p\_k$ (when multiplied by $\partial p\_k$)

Second term: effect of wealth change on demand of good *l* (when
multiplied by $\partial p\_k$) [https://scholar.harvard.edu/files/basilico/files/mwg\_flashcards.pdf]

If you check the symmetry of this matrix (in our case *l* and *k* are *x* and *y*), I believe it should work out.


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