<p>The income effect is defined as <span class="math-container">$\mathbf{-\frac{\partial x_i}{\partial m}x_i}$</span>. </p>

<p>Let <span class="math-container">$x_i$</span> be a normal good; that is, a good whose Marshallian demand increases with an increase in income <span class="math-container">$\left(\frac{\partial x_i}{\partial m} &gt; 0\right)$</span>. Even if there was no substitution effect <span class="math-container">$\left(\frac{\partial h_i}{\partial p_i} = 0\right)$</span> from an increase in own-price, the amount of <span class="math-container">$x_i$</span> consumed would still be reduced as the consumer is effectively poorer. Hence the net income effect in this case is <em>negative</em>, which necessitates the negative sign since <span class="math-container">$x_i$</span> is non-negative. Analogous logic holds if <span class="math-container">$x_i$</span> is an inferior good instead.</p>

<p>This interpretation is consistent with <a href="https://is.muni.cz/el/1431/jaro2014/M9302/um/Nicholson_Micro.pdf" rel="nofollow noreferrer">Nicholson &amp; Snyder (p. 156)</a>.</p>

CC BY-SA 4.0
2018-10-08T03:50:01.610Z
2018-10-08T03:56:02.397Z
2018-10-08T03:56:02.397Z
Kenneth Rios
<p>Recall that the slutsky equation is:</p>

<p><span class="math-container">$$\frac{\partial x_i}{\partial p_i}=\frac{\partial h_i}{\partial p_i}-\frac{\partial x_i}{\partial m}x_i$$</span></p>

<p>I know <span class="math-container">$\frac{\partial h_i}{\partial p_i}$</span> defined as the substitution effect, the second part of the equation is the income effect.</p>

<p>My Question: is the income effect defined as  <span class="math-container">$-\frac{\partial x_i}{\partial m}x_i$</span> or <span class="math-container">$\frac{\partial x_i}{\partial m}x_i$</span> (is the negative sign a defining characteristic?)</p>

<microeconomics><mathematical-economics><slutsky-equation>
Income effect $-\frac{\partial x_i}{\partial m} x_i$ or $\frac{\partial x_i}{\partial m}x_i$?
EconJohn
# Income effect $-\frac{\partial x_i}{\partial m} x_i$ or $\frac{\partial x_i}{\partial m}x_i$?

(Asked by: username_0 on 2018-10-07)

Recall that the slutsky equation is:

$$\frac{\partial x\_i}{\partial p\_i}=\frac{\partial h\_i}{\partial p\_i}-\frac{\partial x\_i}{\partial m}x\_i$$

I know $\frac{\partial h\_i}{\partial p\_i}$ defined as the substitution effect, the second part of the equation is the income effect.

My Question: is the income effect defined as $-\frac{\partial x\_i}{\partial m}x\_i$ or $\frac{\partial x\_i}{\partial m}x\_i$ (is the negative sign a defining characteristic?)

**Tags:** microeconomics, mathematical-economics, slutsky-equation

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## 1 answers

### Answer by username_1 on 2018-10-08. Score: 3 | Accepted answer

The income effect is defined as $\mathbf{-\frac{\partial x\_i}{\partial m}x\_i}$.

Let $x\_i$ be a normal good; that is, a good whose Marshallian demand increases with an increase in income $\left(\frac{\partial x\_i}{\partial m} > 0\right)$. Even if there was no substitution effect $\left(\frac{\partial h\_i}{\partial p\_i} = 0\right)$ from an increase in own-price, the amount of $x\_i$ consumed would still be reduced as the consumer is effectively poorer. Hence the net income effect in this case is *negative*, which necessitates the negative sign since $x\_i$ is non-negative. Analogous logic holds if $x\_i$ is an inferior good instead.

This interpretation is consistent with [Nicholson & Snyder (p. 156)](https://is.muni.cz/el/1431/jaro2014/M9302/um/Nicholson_Micro.pdf).


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