Finding Cobb-Douglas Hicksian Demand using Duality 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.
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).