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I don't understand how to prove slutsky matrix is symmetric for L=2 $$\frac{\partial x_1}{\partial p_2}+\frac{\partial x_1}{\partial w}\cdot x_2= \frac{\partial x_2}{\partial p_1}+\frac{\partial x_2}{\partial w}\cdot x_1$$

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Let $c(p, u)$ be the expenditure function. The Hicksian demand for good $j$ is the derivative of $c$ with respect to $p_j$. $$ \frac{\partial c(p,u)}{\partial p_j} = h_j(p,u). $$ From this, it follows (by Young's theorem) that: $$ \frac{\partial h_j(p,u)}{\partial p_i} = \frac{\partial^2 c(p,u)}{\partial p_j \partial p_i} = \frac{\partial^2 c(p,u)}{\partial p_i \partial p_j} = \frac{\partial h_i(p,u)}{\partial p_j}, $$ So the Hicksian cross price effects are symmetric. Using the Slutsky equation, we get: $$ \begin{align*} &\frac{\partial x_i(p,m)}{\partial p_j} + \frac{\partial x_i(p,m)}{\partial m} x_i(p,m),\\ &= \frac{\partial h_i(p,u)}{\partial p_j},\\ &= \frac{\partial h_j(p,u)}{\partial p_i},\\ &= \frac{\partial x_j(p,m)}{\partial p_i} + \frac{\partial x_j(p,m)}{\partial m} x_j(p,m). \end{align*} $$

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