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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.

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  • $\begingroup$ Maybe could you give us the relationships between both demand functions, and between their slopes wrt price? $\endgroup$
    – Bertrand
    Oct 10 at 18:40
  • $\begingroup$ Income effects can go in both directions. $\endgroup$ Oct 10 at 19:08
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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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