<p>The first term on the RHS of the Slutsky equation (as you write it) should be <span class="math-container">$\frac{\partial h_i(p_i,p_j,I)}{\partial p_j}$</span>, not <span class="math-container">$\frac{\partial h_i(p_i,p_j,I)}{\partial p_i}$</span>.</p>

<p>So your subsequent algebra is fine but you're conflating cross-price substitution with the own-price substitution: when you let <span class="math-container">$i \neq j$</span> in the Slutsky equation, the first term on the RHS <span class="math-container">$\left(\text{again, }\frac{\partial h_i(p_i,p_j,I)}{\partial p_j}\right)$</span> measures the cross-price substitution effect of an increase in the price of <span class="math-container">$j$</span> on the (compensated) demand of <span class="math-container">$i$</span>. You found it to be positive, so goods <span class="math-container">$i$</span> and <span class="math-container">$j$</span> are <em>net</em> substitutes (the cross-price partials of <span class="math-container">$h_i$</span> and <span class="math-container">$h_j$</span> cannot be oppositely signed -- this follows from Shephard's Lemma).</p>

<p>If you let <span class="math-container">$i = j$</span>, however, you will then find that the Slutsky equation implies that the own-price substitution effect is negative for both goods, which you correctly note should be negative:</p>

<p><span class="math-container">$$
\begin{align}
\frac{\partial h_i(p_i,I)}{\partial p_i} &amp;= \frac{\partial x_i}{\partial p_i} + x_i \frac{\partial x_i}{\partial I} \\
&amp;= -\frac{I}{2p_i^2} + \frac{I}{2p_i} \left(\frac{1}{2p_i}\right) \\
&amp;= -\frac{I}{4p_i^2} &lt; 0.
\end{align}
$$</span></p>

CC BY-SA 4.0
2018-10-06T16:15:53.533Z
2018-10-06T16:45:17.083Z
2018-10-06T16:45:17.083Z
Kenneth Rios
<p>We have marshallian demands for goods 1 and 2:</p>

<p><span class="math-container">$x_1^* = \frac{I}{2p_1}$</span>  and <span class="math-container">$x_2^* = \frac{I}{2p_2}$</span>
where <span class="math-container">$I$</span> is income and <span class="math-container">$p_i$</span> is price.</p>

<p>We need to solve the slutsky equation for income effect and substitution effect as such:</p>

<p><span class="math-container">$\frac{D (x_i)}{D (p_j)} = \frac{D(H^i(p1, p2, i))}{ D(p_i)} - x_j*\frac{D(x_i^*)}{D(I)}$</span></p>

<p>So in balance marshallian demand is the same as the compensated demand.</p>

<p>I solved the left hand side of equation and got a result of <span class="math-container">$0$</span>.</p>

<p>Then I moved on to the income effect and got the following result:</p>

<p><span class="math-container">$-\frac{I}{2p_j}*\frac{1}{2p_i} = -\frac{I}{4p_jp_i}$</span></p>

<p>Substituting the results from above in to the slutsky equation gives a positive result for substitution effect. I thought that that the substitution effect is always negative. Can anyone help me out?</p>

<microeconomics>
Slutsky equation with marshallian demand
Khello
# Slutsky equation with marshallian demand

(Asked by: username_0 on 2018-10-06)

We have marshallian demands for goods 1 and 2:

$x\_1^\* = \frac{I}{2p\_1}$ and $x\_2^\* = \frac{I}{2p\_2}$
where $I$ is income and $p\_i$ is price.

We need to solve the slutsky equation for income effect and substitution effect as such:

$\frac{D (x\_i)}{D (p\_j)} = \frac{D(H^i(p1, p2, i))}{ D(p\_i)} - x\_j\*\frac{D(x\_i^\*)}{D(I)}$

So in balance marshallian demand is the same as the compensated demand.

I solved the left hand side of equation and got a result of $0$.

Then I moved on to the income effect and got the following result:

$-\frac{I}{2p\_j}\*\frac{1}{2p\_i} = -\frac{I}{4p\_jp\_i}$

Substituting the results from above in to the slutsky equation gives a positive result for substitution effect. I thought that that the substitution effect is always negative. Can anyone help me out?

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2018-10-06. Score: 3 | Accepted answer

The first term on the RHS of the Slutsky equation (as you write it) should be $\frac{\partial h\_i(p\_i,p\_j,I)}{\partial p\_j}$, not $\frac{\partial h\_i(p\_i,p\_j,I)}{\partial p\_i}$.

So your subsequent algebra is fine but you're conflating cross-price substitution with the own-price substitution: when you let $i \neq j$ in the Slutsky equation, the first term on the RHS $\left(\text{again, }\frac{\partial h\_i(p\_i,p\_j,I)}{\partial p\_j}\right)$ measures the cross-price substitution effect of an increase in the price of $j$ on the (compensated) demand of $i$. You found it to be positive, so goods $i$ and $j$ are *net* substitutes (the cross-price partials of $h\_i$ and $h\_j$ cannot be oppositely signed -- this follows from Shephard's Lemma).

If you let $i = j$, however, you will then find that the Slutsky equation implies that the own-price substitution effect is negative for both goods, which you correctly note should be negative:

$$
\begin{align}
\frac{\partial h\_i(p\_i,I)}{\partial p\_i} &= \frac{\partial x\_i}{\partial p\_i} + x\_i \frac{\partial x\_i}{\partial I} \\
&= -\frac{I}{2p\_i^2} + \frac{I}{2p\_i} \left(\frac{1}{2p\_i}\right) \\
&= -\frac{I}{4p\_i^2} < 0.
\end{align}
$$


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