Why is the partial derivative of $x_1^S(p_1, p_2, \overline{x}_1, \overline{x}_2) \equiv x_1(p_1,p_2,p_1\overline{x}_1+p_2\overline{x}_2)$ for $p_1$ $$ \frac{\partial x_1^S(p_1, p_2, \overline{x}_1, \overline{x}_2)}{\partial p_1}=\frac{\partial x_1(p_1,p_2,\overline{m})}{\partial p_1} +\frac{\partial x_1(p_1,p_2,\overline{m})}{\partial m}\overline{x}_1 $$

where $(\overline{x}_1,\overline{x}_2)$ is the bundle originally demanded at prices $(\overline{p}_1,\overline{p}_2)$ and income $\overline{m}$ and $x_1^S$ is the Slutsky demand function for good 1 and $x_1$ the Marshallian demand function for good 1?

I understand that $m=p_1\overline{x}_1+p_2\overline{x}_2$ because the pivot ensures that the consumer is still having enough income to purchase his old bundle $(\overline{x}_1,\overline{x}_2)$.

Chapter 8 of Varian shows this to derive the Slutsky equation using calculus. I don't understand the right side of the equation.

I know that $p_1\overline{x}_1+p_2\overline{x}_2=m$ and $\overline{p}_1\overline{x}_1+\overline{p}_2\overline{x}_2=\overline{m}$ where $m$ is the new budget.


1 Answer 1


This is more of a calculus question. Recall the total differential of a function $f(z_1,z_2,z_3)$:
\begin{equation} \mathrm df(z_1,z_2,z_3)=\frac{\partial f(z_1,z_2,z_3)}{\partial z_1}\mathrm dz_1 + \frac{\partial f(z_1,z_2,z_3)}{\partial z_2}\mathrm dz_2+\frac{\partial f(z_1,z_2,z_3)}{\partial z_3}\mathrm dz_3. \end{equation} Hence, \begin{equation} \frac{\mathrm df(z_1,z_2,z_3)}{\mathrm dz_1}=\frac{\partial f(z_1,z_2,z_3)}{\partial z_1}\frac{\mathrm dz_1}{\mathrm dz_1} + \frac{\partial f(z_1,z_2,z_3)}{\partial z_2}\frac{\mathrm dz_2}{\mathrm dz_1} + \frac{\partial f(z_1,z_2,z_3)}{\partial z_3}\frac{\mathrm dz_3}{\mathrm dz_1} \end{equation} Applying to your case, we substitute $p_1$ for $z_1$, $p_2$ for $z_2$, and $\overline m$ for $z_3$ (noting that $\overline m(p_1,p_2)$ is a function of the prices):
\begin{align}\require{cancel} \frac{\mathrm dx_1(p_1,p_2,\overline m)}{\mathrm dp_1}&=\frac{\partial x_1(p_1,p_2,\overline m)}{\partial p_1}\cancelto{1}{\frac{\mathrm dp_1}{\mathrm dp_1}} + \frac{\partial x_1(p_1,p_2,\overline m)}{\partial p_2}\cancelto{0}{\frac{\mathrm dp_2}{\mathrm dp_1}} + \frac{\partial x_1(p_1,p_2,\overline m)}{\partial \overline m}\frac{\mathrm d\overline m}{\mathrm dp_1}\\ &=\frac{\partial x_1(p_1,p_2,\overline m)}{\partial p_1} + \frac{\partial x_1(p_1,p_2,\overline m)}{\partial \overline m}\overline x_1 \end{align}

  • $\begingroup$ Sorry, I don't quite understand tbh. I think you mean "substitute $p_1$ for $z_1$"? Also, why does the total differential not include the partial derivative for $p_2$ but only for $p_1$ and $m$? And why is $p_1\overline{x}_1+p_2\overline{x}_2=\overline{m}$? Shouldn't this be m if $\overline{m}$ is the original budget and $p_1, p_2$ are the new prices? @Herr K. $\endgroup$ Commented May 18, 2021 at 14:00
  • $\begingroup$ Got it, nvm, it took a while but it finally made "pling" up there. $\endgroup$ Commented May 18, 2021 at 17:07
  • 1
    $\begingroup$ @j3141592653589793238: I edited in some intermediate steps. Hopefully things will be clearer. Regarding $\overline m$, I took that from the first identity in your post: $x_1^S(p_1, p_2, \overline{x}_1, \overline{x}_2) \equiv x_1(p_1,p_2,p_1\overline{x}_1+p_2\overline{x}_2)$, which defines the third argument of $x_1$ to be $p_1\overline{x}_1+p_2\overline{x}_2$ (whether you call this $\overline m$ or otherwise). $\endgroup$
    – Herr K.
    Commented May 18, 2021 at 17:20

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