Derivative of $x_1^S(p_1, p_2, \overline{x}_1, \overline{x}_2) \equiv x_1(p_1,p_2,p_1\overline{x}_1 + p_2\overline{x}_2)$ to derive Slutsky equation

Question

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.

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}