<p>Given <span class="math-container">$x_{m}$</span> in Slutsky form</p>
<p><span class="math-container">$$
\frac{\partial x_{m}(p_{x},p_{y},I)}{\partial p_{x}} = \frac{\partial x_{m}}{\partial p_{x}} \rfloor_{U=constant} - x_{c} \frac{\partial x_{m}}{\partial p_{x}} \\
= \text{substitution effect} + \text{income effect}
$$</span></p>
<p>From Nicholson and Snyder (Microeconomic Theory 12th Ed.) (p. 158)
&quot;The sign of the income effect <span class="math-container">$-x_{c}\partial x/\partial I$</span> depends on the sign of <span class="math-container">$\partial x/\partial I$</span>. If <span class="math-container">$x$</span> is a normal good, then <span class="math-container">$\partial x/\partial I$</span> is positive and the entire income effect, like the substitution effect, is negative...In the case of an inferior good, <span class="math-container">$\partial x/\partial I &lt; 0$</span> and the two terms have different signs. Hence, the overall impact of a change in the price of a good is ambiguous - it all depends on the relative sizes of the effects. It is at least theoretically possible that, in the inferior good case, the second term could dominate the first, leading to Giffen's paradox...&quot;</p>

CC BY-SA 4.0
2021-10-11T01:23:24.503Z
2021-10-11T01:23:24.503Z
kleinerde
<p>The compensated demand curve eliminates income effects. It reflects only substitution effects. Given that the Marshallian demand curve reflects income effects, doesn't this mean it is always more elastic than the Hicksian, because the quantity is more sensitive to price, and therefore always shallower?</p>
<p>We know:</p>
<p><span class="math-container">$$
x_{c}(p_{x},p_{y},U) = x_{m}(p_{x},p_{y},E(p_{x},p_{y},U))
$$</span>
and
<span class="math-container">$$
\frac{\partial x_{c}}{\partial p_{x}} = \frac{\partial x_{m}}{\partial p_{x}} + \frac{\partial x_{m}}{\partial E}\frac{\partial E}{\partial p_{x}}
$$</span>
Where <span class="math-container">$E$</span> is the expenditure function <span class="math-container">$E(p_{x},p_{y},U)$</span> and <span class="math-container">$U$</span> is utility.</p>
<p>The second term represents the removal of the income effect, i.e.
<span class="math-container">$
-\frac{\partial x_{m}}{\partial E}\frac{\partial E}{\partial p_{x}}
$</span>.</p>
<p>I'm now wondering if the term <span class="math-container">$\frac{\partial x_{m}}{\partial E}$</span> which is equal to <span class="math-container">$\frac{\partial x_{m}}{\partial I}$</span> under the assumption all income is expended is different for normal and inferior goods and if this effect always reverses the steepness relationship.</p>

CC BY-SA 4.0
2021-10-10T18:40:41.287Z
Maybe could you give us the relationships between both demand functions, and between their slopes wrt price?
Bertrand
CC BY-SA 4.0
2021-10-10T19:08:23.277Z
Income effects can go in both directions.
Michael Greinecker
<demand><income-effect>
Is Hicksian always steeper than Marshallian?
kleinerde
# Is Hicksian always steeper than Marshallian?

(Asked by: username_0 on 2021-10-10)

The compensated demand curve eliminates income effects. It reflects only substitution effects. Given that the Marshallian demand curve reflects income effects, doesn't this mean it is always more elastic than the Hicksian, because the quantity is more sensitive to price, and therefore always shallower?

We know:

$$
x\_{c}(p\_{x},p\_{y},U) = x\_{m}(p\_{x},p\_{y},E(p\_{x},p\_{y},U))
$$
and
$$
\frac{\partial x\_{c}}{\partial p\_{x}} = \frac{\partial x\_{m}}{\partial p\_{x}} + \frac{\partial x\_{m}}{\partial E}\frac{\partial E}{\partial p\_{x}}
$$
Where $E$ is the expenditure function $E(p\_{x},p\_{y},U)$ and $U$ is utility.

The second term represents the removal of the income effect, i.e.
$
-\frac{\partial x\_{m}}{\partial E}\frac{\partial E}{\partial p\_{x}}
$.

I'm now wondering if the term $\frac{\partial x\_{m}}{\partial E}$ which is equal to $\frac{\partial x\_{m}}{\partial I}$ under the assumption all income is expended is different for normal and inferior goods and if this effect always reverses the steepness relationship.

**Tags:** demand, income-effect

> **Comments:**
> **username_1**: Maybe could you give us the relationships between both demand functions, and between their slopes wrt price?  
> **username_2**: Income effects can go in both directions.  

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## 1 answers

### Answer by username_0 on 2021-10-11. Score: 1

Given $x\_{m}$ in Slutsky form

$$
\frac{\partial x\_{m}(p\_{x},p\_{y},I)}{\partial p\_{x}} = \frac{\partial x\_{m}}{\partial p\_{x}} \rfloor\_{U=constant} - x\_{c} \frac{\partial x\_{m}}{\partial p\_{x}} \\
= \text{substitution effect} + \text{income effect}
$$

From Nicholson and Snyder (Microeconomic Theory 12th Ed.) (p. 158)
"The sign of the income effect $-x\_{c}\partial x/\partial I$ depends on the sign of $\partial x/\partial I$. If $x$ is a normal good, then $\partial x/\partial I$ is positive and the entire income effect, like the substitution effect, is negative...In the case of an inferior good, $\partial x/\partial I < 0$ and the two terms have different signs. Hence, the overall impact of a change in the price of a good is ambiguous - it all depends on the relative sizes of the effects. It is at least theoretically possible that, in the inferior good case, the second term could dominate the first, leading to Giffen's paradox..."


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