<p>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: $\frac{\partial e(\textbf{p},u)}{\partial p_i} = h_i(\textbf{p},u)).$ </p>

<p>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.</p>

CC BY-SA 4.0
2018-09-11T16:39:31.720Z
2018-09-11T16:52:57.410Z
2018-09-11T16:52:57.410Z
superhulk
<p>Given the indirect utility function:
<em>V={M^2}/{4P1P2}</em>,
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. </p>

<microeconomics><consumer-theory><preferences>
Slutsky Decomposition from Indirect Utility Function
S.Rana
# Slutsky Decomposition from Indirect Utility Function

(Asked by: username_0 on 2018-09-11)

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.

**Tags:** microeconomics, consumer-theory, preferences

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## 1 answers

### Answer by username_1 on 2018-09-11. Score: 0 | Accepted answer

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.


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