We have a Slutsky matrix:
\begin{bmatrix} \partial x_{1}^H/\partial P_1 & \partial x_{1}^H/\partial P_2 & \dots & \partial x_{1}^H/\partial P_n \\ \partial x_{2}^H/\partial P_1 & \partial x_{2}^H/\partial P_2 & \dots & \partial x_{2}^H/\partial P_n \\ \vdots & \vdots & \ddots & \vdots \\ \partial x_{n}^H/\partial P_1 & \partial x_{n}^H/\partial P_2 & \dots & \partial x_{n}^H/\partial P_n \end{bmatrix}
Where $x_i^H$ is a Hicksian demand for good $i$. These demands are then differentiated by prices.
The interpretation of this matrix should be related to substitutes/complements.
Goods are substitutes: If $\partial x_{i}^H/\partial P_j > 0$
Goods are complements: If $\partial x_{i}^H/\partial P_j < 0$
However, I would like to identify the Slutsky matrix in case of CES function:
$$U = \left( x_1^{\rho} + x_2^{\rho} \right)^{\frac{1}{\rho}}$$
I solve the expenditure minimization problem:
$$\mathscr{L} = P_1 x_1 + P_2 x_2 + \lambda \left(\widetilde{U} - \left( x_1^{\rho} + x_2^{\rho} \right)^{\frac{1}{\rho}} \right)$$
The hicksian demands should be the following:
$$x_1^H = \frac{\widetilde{U}}{\left( \left( \frac{P_2}{P_1} \right)^\frac{\rho}{\rho-1}+1 \right)^\frac{1}{\rho}}$$
$$x_2^H = \frac{\widetilde{U}}{\left( \left( \frac{P_1}{P_2} \right)^\frac{\rho}{\rho-1}+1 \right)^\frac{1}{\rho}}$$
If I differentiate $x_2^H$ by $P_1$ I get the following:
$$\frac{\partial x_2^H}{\partial P_1} = - \frac{1}{\rho} \frac{\rho}{\rho-1} \cdot \widetilde{U} \left( \left( \frac{P_1}{P_2} \right)^\frac{\rho}{\rho-1} +1 \right)^{-\frac{1}{\rho}-1} P_1^{\frac{\rho}{\rho-1}-1} P_2^{-\frac{\rho}{\rho-1}}$$
What concerns $P_1$, $P_2$ and $\widetilde{U}$, they are all positive, so they are not that interesting, simplifying the result into:
$$\frac{\partial x_2^H}{\partial P_1} = - \frac{1}{\rho-1} \cdot K$$
And now, this is weird... We can see, that this element would be always positive since $\rho \in (-\infty; 1)$. This would tell us the goods would be substitutes no matter what...
However, we know that in CES function, if $\rho > 0$ goods are substitutes, if $\rho < 0$ goods are complements. Therefore, I would expect that signs of the elements in Slutsky matrix should depend on $\rho$!
- Why does this not hold?
- Why is the sign of $\partial x_2^H / \partial P_1$ independent on $\rho$?
