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In my 3rd year Microeconomics course we're deriving the Slutsky equation, and we have this general form at the start of the derivation:

$$x _ { l } ( p , e ( p , u ) ) = h _ { l } ( p , u )$$

And:

$$e ( p , u ) = y$$

Taking the total differential:

$$\biggr( \frac { \partial x _ { l } } { \partial p _ { l } } + \frac { \partial x _ { l } } { \partial y } \cdot \frac { \partial e } { \partial p _ { l } } \biggr) \mathrm dp_l = \frac { \partial h_l}{ \partial p_l}\cdot \mathrm dp_l $$

I'm unfamiliar with total differentials, though my understanding is it involves taking the derivative WRT all variables.

Could someone please advise:

  • why $\partial$ and $\mathrm d$ are being used simultaneously?

  • why the addition symbol is being used to split up terms on the LHS

  • confirmation of whether the chain rule is involved

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For the same reason that we can write for $f(x,y,z)$ - $$df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz$$ Also, the chain rule has been used as - $$dx_l = \frac{\partial x_l}{\partial p_l}dp_l + \frac{\partial x_l}{\partial y}dy $$ $$\implies dx_l = \frac{\partial x_l}{\partial p_l}dp_l + \frac{\partial x_l}{\partial y}\frac{\partial e}{\partial p_l}dp_l$$ (as $e=y$)

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  • $\begingroup$ That's excellent. I still don't grasp it fully (as my math is super shaky still), though intuitively it makes a bit more sense. Thanks so much! $\endgroup$ – altec May 26 at 3:11

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