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We define the change in demand due to income effect as (assume there are 2 goods and the price of only one of them, say $p_1$, changes)

$dx_1^N = x_1(p_1 + dp_1, p_2, m) - x_1(p_1 + dp_1, p_2, m + dm)$.

The total change in quantity demanded is supposed to be

$dx_1 = dx_1^S + dx_1^N$

However, for the Slutsky equation, we adopt the convention

$dx_1 = dx_1^S + dx_1^M$, where $dx_1^M = -dx_1^N$.

My first question - why exactly do we adopt this convention?

Now assuming that we start off with an endowment $(\omega_1, \omega_2)$, the Slutsky equation has to be modified to accommodate the endowment income effect, because a change in price will necessarily lead to a change in income. In the Intermediate Microeconomics book by Varian, the rate of change in demand due to endowment income effect is defined as

$\frac{\partial x_1^M}{\partial m} \frac{\partial m}{\partial p_1}$, and so the Slutsky equation becomes (in terms of rate of change w.r.t price)

$\frac{\partial x_1}{\partial p_1} = \frac{\partial x_1^S}{\partial p_1} - \frac{\partial x_1^M}{\partial p_1} + \frac{\partial x_1^M}{\partial m} \frac{\partial m}{\partial p_1}$

Why can't we define the endowment income effect as $\frac{\partial x_1^N}{\partial m} \frac{\partial m}{\partial p_1} = - \frac{\partial x_1^M}{\partial m} \frac{\partial m}{\partial p_1}$?

A mathematically rigorous answer would be appreciated.

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