<p>You have found the value function <span class="math-container">$V(p,I)$</span> which in your case is given as</p>
<p><span class="math-container">$$V(p,I) = \alpha_1^{\alpha_1}\alpha_2^{\alpha_2}\frac{I}{p_1^{\alpha_1}p_2^{\alpha_2}},$$</span></p>
<p>where <span class="math-container">$\alpha_2 = 1- \alpha_1$</span> and <span class="math-container">$I$</span> is income.</p>
<p>The Hicks demand is a function of the utility level denoted <span class="math-container">$u$</span>. An agent facing prices <span class="math-container">$p$</span> and income <span class="math-container">$I$</span> gets utility <span class="math-container">$V(p,I) = u$</span>. Therefore you can set</p>
<p><span class="math-container">$$u = \alpha_1^{\alpha_1}\alpha_2^{\alpha_2}\frac{I}{p_1^{\alpha_1}p_2^{\alpha_2}},$$</span></p>
<p>and invert to solve for <span class="math-container">$I$</span> in order to get</p>
<p><span class="math-container">$$\frac{p_1^{\alpha_1}p_2^{\alpha_2}}{\alpha_1^{\alpha_1}\alpha_2^{\alpha_2}}u = I,$$</span>
which is the expenditure function. You can insert this in your Marshall demand</p>
<p><span class="math-container">$$\frac{\alpha_j I}{p_j} = \left(\frac{\alpha_j}{p_j}\right)\frac{p_1^{\alpha_1}p_2^{\alpha_2}}  {\alpha_1^{\alpha_1}\alpha_2^{\alpha_2}}u = h^\star(p,u).$$</span></p>
<p>Or in the case of <span class="math-container">$J$</span> goods: You have CD preferences with well known Marshall demand</p>
<p><span class="math-container">$$(1)\ \  x^\star_j(p,I) = \frac{\alpha_j I}{p_j} \Leftrightarrow \frac{x_j}{\alpha_j} = \frac{I}{p_j},$$</span></p>
<p>and the utility function is</p>
<p><span class="math-container">$$(2) \ \ U(x) = \prod_j x_j^{\alpha_j} = A \prod_j \left(\frac{x_j}{\alpha_j} \right)^{\alpha_j},$$</span>
where <span class="math-container">$A := \prod_j \alpha_j^{\alpha_j}$</span> the constant <span class="math-container">$A$</span> serves the purpose of making the algebra somewhat easier. Inserting from (1) in (2) gives you value function</p>
<p><span class="math-container">$$V(p,I) = A \prod_j \left(\frac{I}{p_j} \right)^{\alpha_j} = \frac{AI}{\bar p},$$</span>
where <span class="math-container">$\bar p:= \prod_j p_j^{\alpha_j}$</span> and <span class="math-container">$I$</span> is constant across goods hence not bound by index so when <span class="math-container">$\sum_j \alpha_j = 1$</span> it <span class="math-container">$\prod_j I^{\alpha_j} = I$</span>. From here you isolate income to get</p>
<p><span class="math-container">$$\frac{\bar p}{A}V(p,I) = I$$</span></p>
<p>you then insert this in Marshall demand to get</p>
<p><span class="math-container">$$x^\star(p,I) = x^\star\left(p,\frac{\bar p V(p,I)}{A}\right) = h^\star(p,V(p,I)).$$</span></p>
<p>Knowing the Marshall demand <span class="math-container">$\alpha_j I/p_j$</span> insert <span class="math-container">$\bar p V(p,I)/A$</span> with <span class="math-container">$V(p,I)$</span>to get</p>
<p><span class="math-container">$$h^\star(p,V(p,I)) = \frac{\alpha_j}{p_j} \frac{\bar p V(p,I)}{A} ,$$</span></p>
<p>let <span class="math-container">$V(p,I) = u$</span> to get</p>
<p><span class="math-container">$$h^\star(p,u) = \frac{\alpha_j}{p_j} \frac{\bar p u}{A} = \frac{\alpha_j}{p_j} E(p,u),$$</span></p>
<p>where <span class="math-container">$E(p,u)= \frac{\bar p u}{A}$</span> is expenditure function (this follows from the fact that we arrived at <span class="math-container">$\bar p u/A$</span> as an expression for income by inverting the value function).</p>

CC BY-SA 4.0
2020-12-27T20:58:48.997Z
@CorporateNationalism you should accept this answer so that other people looking for this same answer can more easily find it, it also helps get the site off of beta :)
Brennan
CC BY-SA 4.0
2020-12-18T22:38:05.237Z
2020-12-19T01:36:52.287Z
2020-12-19T01:36:52.287Z
Jesper Hybel
<p>I prefer another approach. I use duality to find the <em>cost function</em> and then use Shepard's Lemma to derive the Hicksian demands. I will demonstrate below for the simple 2 good case, rather than generally, for simplicity.</p>
<p>Begin with the indirect utility function:
<span class="math-container">$$V(p,I) = \alpha_1^{\alpha_1}\alpha_2^{\alpha_2}\frac{I}{p_1^{\alpha_1}p_2^{\alpha_2}} = \left(\frac{\alpha_1}{p_1}\right)^{\alpha_1}\left(\frac{\alpha_2}{p_2}\right)^{\alpha_2}I$$</span>
and we use duality to impose that <span class="math-container">$V(p,I)=u$</span> and <span class="math-container">$C(p,u) = I$</span> at the optimal value of these functions (the solution to the utility maximization problem is the same as that of the cost minimization problem, you can explore this concept on your own time).</p>
<p>Then we can solve for <span class="math-container">$C(p,u)$</span> by substituting the above relationships in:
<span class="math-container">$$u=\left(\frac{\alpha_1}{p_1}\right)^{\alpha_1}\left(\frac{\alpha_2}{p_2}\right)^{\alpha_2}C(p,u)$$</span>
then let <span class="math-container">$A:=\left(\frac{\alpha_1}{p_1}\right)^{\alpha_1}\left(\frac{\alpha_2}{p_2}\right)^{\alpha_2}$</span> for simplicity, the above can be represented as <span class="math-container">$u=AC(p,u)$</span> which implies that the cost function is <span class="math-container">$C(p,u) = A^{-1}u$</span>, therefore
<span class="math-container">$$C(p,u) = \left(\frac{p_1}{\alpha_1}\right)^{\alpha_1}\left(\frac{p_2}{\alpha_2}\right)^{\alpha_2}u$$</span></p>
<p>Now that we have the form of the cost function, we can invoke Shepard's Lemma which states that the derivative of the cost function with respect to the price of a good gives us the Hicksian demand for said good. Think of it in terms of the budget constraint: <span class="math-container">$p_{1}x_1 + p_{2}x_2 \leq I$</span>, where <span class="math-container">$I$</span> would be our expenditure should this constraint be binding. If we have optimal values for <span class="math-container">$x_i$</span>, then taking a derivative of the constraint/expenditure with respect to, say, <span class="math-container">$p_1$</span> should give us <span class="math-container">$x_{1}^*$</span> right?</p>
<p>Shepard's Lemma can be written succinctly as
<span class="math-container">$$\frac{\partial c(p,u)}{\partial p_i} = h_{i}(p,u)$$</span></p>
<p>So, for the Hicksian demand of good 1, for example, it would be
<span class="math-container">$$h_{1}(p,u) = \frac{\partial C(p,u)}{\partial p_{1}} = \frac{1}{\alpha_1}\alpha_{1}\left(\frac{p_1}{\alpha_1}\right)^{\alpha_1-1}\left(\frac{p_2}{\alpha_2}\right)^{\alpha_2}u = \left(\frac{p_1}{\alpha_1}\right)^{\alpha_1-1}\left(\frac{p_2}{\alpha_2}\right)^{\alpha_2}u$$</span></p>

CC BY-SA 4.0
2020-12-27T21:24:22.197Z
2020-12-27T21:24:22.197Z
Brennan
<p>I'm trying to follow the in-text examples from Mas-Colell. I can confirm I have the correct first order-conditions and hence the Marshallian demand functions for Example 3.D.1: <span class="math-container">$u(x_1,x_2) = x_1^\alpha x_2^{1-\alpha}$</span> subject to the budget constraint, <span class="math-container">$p_1x_1 + p_2x_2 =w$</span>. By going through the example and taking Lagrange, I found the Marshallian demands to be <span class="math-container">$x_1(p,w)=\frac{\alpha w}{p_1}$</span> and <span class="math-container">$x_2(p,w)=\frac{(1-\alpha) w}{p_2}$</span>. Fine.</p>
<p>The part I'm having difficulty with is finding the Hicksian demand, we know from 3.E.4 that <span class="math-container">$x(p,w)=h(p,v(p,w))$</span>, where <span class="math-container">$v(p,w)$</span> is indirect utility, it's made clear that we can find indirect utility by substituting our Marshallian demand functions into <span class="math-container">$u(p,w)$</span>.</p>
<p>I attempt to find v(p,w) by substituting the Marshallian demand for <span class="math-container">$x_2$</span> into the <span class="math-container">$u(x_1,x_2)$</span> : V(p,w)=
<span class="math-container">$(\frac{\alpha w}{p_1})^\alpha [ \frac{(1-\alpha)m}{p_2} ] ^{1-\alpha}$</span></p>
<p>To quote the notes given to me by my tutor, 'Marshallian is Hicksian if we substitute utility by indirect tility function:</p>
<p><span class="math-container">$x^*(p,w)=h^*(p,V(p,w))$</span></p>
<p>I've reached a bit of a dead end here, I have the indirect utility function, but don't know how to get the Hicksian demand.</p>

CC BY-SA 4.0
2020-12-18T15:16:26.783Z
When you have the value function $V(p,w) = k \cdot w$ - which in this case is simply a constant $k(p,\alpha)$ multiplied by income - invert to solve for income $w = V(p,I) /k$ then insert this expression in Marshall demand to get $h(p,V) = x(p,V/k)$.
Jesper Hybel
CC BY-SA 4.0
2020-12-18T16:23:43.197Z
TYPO: the »I« as argument in V should be w.
Jesper Hybel
CC BY-SA 4.0
2020-12-19T14:02:30.303Z
You could consider accepting the answer or entering a dialogue about what you think is missing.
Jesper Hybel
<microeconomics><walrasian><mwg><duality>
Finding Cobb-Douglas Hicksian Demand using Duality
CorporateNationalism
# Finding Cobb-Douglas Hicksian Demand using Duality

(Asked by: username_0 on 2020-12-18)

I'm trying to follow the in-text examples from Mas-Colell. I can confirm I have the correct first order-conditions and hence the Marshallian demand functions for Example 3.D.1: $u(x\_1,x\_2) = x\_1^\alpha x\_2^{1-\alpha}$ subject to the budget constraint, $p\_1x\_1 + p\_2x\_2 =w$. By going through the example and taking Lagrange, I found the Marshallian demands to be $x\_1(p,w)=\frac{\alpha w}{p\_1}$ and $x\_2(p,w)=\frac{(1-\alpha) w}{p\_2}$. Fine.

The part I'm having difficulty with is finding the Hicksian demand, we know from 3.E.4 that $x(p,w)=h(p,v(p,w))$, where $v(p,w)$ is indirect utility, it's made clear that we can find indirect utility by substituting our Marshallian demand functions into $u(p,w)$.

I attempt to find v(p,w) by substituting the Marshallian demand for $x\_2$ into the $u(x\_1,x\_2)$ : V(p,w)=
$(\frac{\alpha w}{p\_1})^\alpha [ \frac{(1-\alpha)m}{p\_2} ] ^{1-\alpha}$

To quote the notes given to me by my tutor, 'Marshallian is Hicksian if we substitute utility by indirect tility function:

$x^\*(p,w)=h^\*(p,V(p,w))$

I've reached a bit of a dead end here, I have the indirect utility function, but don't know how to get the Hicksian demand.

**Tags:** microeconomics, walrasian, mwg, duality

> **Comments:**
> **username_1**: When you have the value function $V(p,w) = k \cdot w$ - which in this case is simply a constant $k(p,\alpha)$ multiplied by income - invert to solve for income $w = V(p,I) /k$ then insert this expression in Marshall demand to get $h(p,V) = x(p,V/k)$.  
> **username_1**: TYPO: the »I« as argument in V should be w.  
> **username_1**: You could consider accepting the answer or entering a dialogue about what you think is missing.  

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## 2 answers

### Answer by username_1 on 2020-12-18. Score: 6

You have found the value function $V(p,I)$ which in your case is given as

$$V(p,I) = \alpha\_1^{\alpha\_1}\alpha\_2^{\alpha\_2}\frac{I}{p\_1^{\alpha\_1}p\_2^{\alpha\_2}},$$

where $\alpha\_2 = 1- \alpha\_1$ and $I$ is income.

The Hicks demand is a function of the utility level denoted $u$. An agent facing prices $p$ and income $I$ gets utility $V(p,I) = u$. Therefore you can set

$$u = \alpha\_1^{\alpha\_1}\alpha\_2^{\alpha\_2}\frac{I}{p\_1^{\alpha\_1}p\_2^{\alpha\_2}},$$

and invert to solve for $I$ in order to get

$$\frac{p\_1^{\alpha\_1}p\_2^{\alpha\_2}}{\alpha\_1^{\alpha\_1}\alpha\_2^{\alpha\_2}}u = I,$$
which is the expenditure function. You can insert this in your Marshall demand

$$\frac{\alpha\_j I}{p\_j} = \left(\frac{\alpha\_j}{p\_j}\right)\frac{p\_1^{\alpha\_1}p\_2^{\alpha\_2}} {\alpha\_1^{\alpha\_1}\alpha\_2^{\alpha\_2}}u = h^\star(p,u).$$

Or in the case of $J$ goods: You have CD preferences with well known Marshall demand

$$(1)\ \ x^\star\_j(p,I) = \frac{\alpha\_j I}{p\_j} \Leftrightarrow \frac{x\_j}{\alpha\_j} = \frac{I}{p\_j},$$

and the utility function is

$$(2) \ \ U(x) = \prod\_j x\_j^{\alpha\_j} = A \prod\_j \left(\frac{x\_j}{\alpha\_j} \right)^{\alpha\_j},$$
where $A := \prod\_j \alpha\_j^{\alpha\_j}$ the constant $A$ serves the purpose of making the algebra somewhat easier. Inserting from (1) in (2) gives you value function

$$V(p,I) = A \prod\_j \left(\frac{I}{p\_j} \right)^{\alpha\_j} = \frac{AI}{\bar p},$$
where $\bar p:= \prod\_j p\_j^{\alpha\_j}$ and $I$ is constant across goods hence not bound by index so when $\sum\_j \alpha\_j = 1$ it $\prod\_j I^{\alpha\_j} = I$. From here you isolate income to get

$$\frac{\bar p}{A}V(p,I) = I$$

you then insert this in Marshall demand to get

$$x^\star(p,I) = x^\star\left(p,\frac{\bar p V(p,I)}{A}\right) = h^\star(p,V(p,I)).$$

Knowing the Marshall demand $\alpha\_j I/p\_j$ insert $\bar p V(p,I)/A$ with $V(p,I)$to get

$$h^\star(p,V(p,I)) = \frac{\alpha\_j}{p\_j} \frac{\bar p V(p,I)}{A} ,$$

let $V(p,I) = u$ to get

$$h^\star(p,u) = \frac{\alpha\_j}{p\_j} \frac{\bar p u}{A} = \frac{\alpha\_j}{p\_j} E(p,u),$$

where $E(p,u)= \frac{\bar p u}{A}$ is expenditure function (this follows from the fact that we arrived at $\bar p u/A$ as an expression for income by inverting the value function).

> **Comments:**
> **username_2**: @username_0 you should accept this answer so that other people looking for this same answer can more easily find it, it also helps get the site off of beta :)  

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### Answer by username_2 on 2020-12-27. Score: 1

I prefer another approach. I use duality to find the *cost function* and then use Shepard's Lemma to derive the Hicksian demands. I will demonstrate below for the simple 2 good case, rather than generally, for simplicity.

Begin with the indirect utility function:
$$V(p,I) = \alpha\_1^{\alpha\_1}\alpha\_2^{\alpha\_2}\frac{I}{p\_1^{\alpha\_1}p\_2^{\alpha\_2}} = \left(\frac{\alpha\_1}{p\_1}\right)^{\alpha\_1}\left(\frac{\alpha\_2}{p\_2}\right)^{\alpha\_2}I$$
and we use duality to impose that $V(p,I)=u$ and $C(p,u) = I$ at the optimal value of these functions (the solution to the utility maximization problem is the same as that of the cost minimization problem, you can explore this concept on your own time).

Then we can solve for $C(p,u)$ by substituting the above relationships in:
$$u=\left(\frac{\alpha\_1}{p\_1}\right)^{\alpha\_1}\left(\frac{\alpha\_2}{p\_2}\right)^{\alpha\_2}C(p,u)$$
then let $A:=\left(\frac{\alpha\_1}{p\_1}\right)^{\alpha\_1}\left(\frac{\alpha\_2}{p\_2}\right)^{\alpha\_2}$ for simplicity, the above can be represented as $u=AC(p,u)$ which implies that the cost function is $C(p,u) = A^{-1}u$, therefore
$$C(p,u) = \left(\frac{p\_1}{\alpha\_1}\right)^{\alpha\_1}\left(\frac{p\_2}{\alpha\_2}\right)^{\alpha\_2}u$$

Now that we have the form of the cost function, we can invoke Shepard's Lemma which states that the derivative of the cost function with respect to the price of a good gives us the Hicksian demand for said good. Think of it in terms of the budget constraint: $p\_{1}x\_1 + p\_{2}x\_2 \leq I$, where $I$ would be our expenditure should this constraint be binding. If we have optimal values for $x\_i$, then taking a derivative of the constraint/expenditure with respect to, say, $p\_1$ should give us $x\_{1}^\*$ right?

Shepard's Lemma can be written succinctly as
$$\frac{\partial c(p,u)}{\partial p\_i} = h\_{i}(p,u)$$

So, for the Hicksian demand of good 1, for example, it would be
$$h\_{1}(p,u) = \frac{\partial C(p,u)}{\partial p\_{1}} = \frac{1}{\alpha\_1}\alpha\_{1}\left(\frac{p\_1}{\alpha\_1}\right)^{\alpha\_1-1}\left(\frac{p\_2}{\alpha\_2}\right)^{\alpha\_2}u = \left(\frac{p\_1}{\alpha\_1}\right)^{\alpha\_1-1}\left(\frac{p\_2}{\alpha\_2}\right)^{\alpha\_2}u$$


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