# Slutsky Decomposition from Indirect Utility Function [closed]

Given the indirect utility function: V={M^2}/{4P1P2}, how do we establish the Slutsky Decomposition? I used Roy's Identity to get the Demand, but I'm stuck with the other components of the Slutsky Equation.

## closed as off-topic by Giskard, Kitsune Cavalry♦Sep 15 '18 at 23:07

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Since you already know the demand correspondences, you can use use the duality principle as follows: $v(\textbf{p},e(\textbf{p},u)) = u.$ Using this we get $\frac{4e(\textbf{p},u)}{p_1p_2}=u$, or $e(\textbf{p},u)) = \frac{up_1p_2}{4}.$ Now, using Shephard's Lemma, we can obtain the Hicksian demands as: $\frac{\partial e(\textbf{p},u)}{\partial p_i} = h_i(\textbf{p},u)).$
However, the job is not yet done. You might get stuck with the $u$ while evaluating the Slutsky's equation. In other words, you need to find the utility function as well. The demand correspondences are(I hope you've obtained the same expressions) $x_1(p_1,p_2,m)=m/2p_1$ and $x_2(p_1,p_2,m)=m/2p_2$. Since the indirect utility function is homogenous of degree zero, you can further write it as $v(p_1,1,m)=v(p,m)=m^2/4p=(m/2).(m/2p).$ From the demand correspondences obtained, you have to represent $m/2$ and $m/2p$ in terms of $x_1$ and $x_2$ ($p_2=1$, as a consequence of homogeneity of $v(p_1,p_2,m)$), and substitue it back into the indirect utility function. Thus, we obtain the the utility function $U(x_1,x_2)=x_1.x_2$(since we obtain the indirect utility function by substituting the walrasian demands into the given utility function, backtracking the same steps gives us back the utility from the indirect utility). I think now one can easily evaluate the Slutsky's Equation.