The indirect utility function describes the maximum amount of utility a consumer can achieve, given a fixed amount of income and facing fixed prices. For two goods it is defined as

$V(P_1,P_2,M) = U(X_1^*,X_2^*)$

where $V$ denotes the indirect utility function, $P_1$ is the price of good 1, $P_2$ is the price of good 2, $M$ is the consumers income, $U$ is the utility function, $X_1^*$ is the quantity of good 1 consumed in equilibrium and $X_2^*$ is the quantity of good 2 consumed in equilibrium.

Cobb Douglas Utility function

Consider the scenario where a consumer has the utility function described by Cobb Douglas Preferences, such that

$U = X_1^\alpha X_2^{(1-\alpha)}$

and has the following budget constraint:

$M = P_1 X_1 + P_2 X_2$

It is important to note that $X_1$ and $X_2$ are just feasible options for good 1 and 2 and are not the equilibrium level of consumption $X_1^*$ and $X_2^*$.

To calculate the indirect utility function, we need to first find the demand equations $X_1^*$ and $X_2^*$. Using our equilibrium condition

$\frac{MU \text{ of } X_1}{P_1}=\frac{MU\text{ of  }X_2}{P_1}$

And substituting in the marginal utility for both good $X_1$ and good $X_2$

$MU \text{ of } X_1 = \alpha \frac{U}{X_1}$

$MU \text{ of } X_2 = (1-\alpha) \frac{U}{X_2}$

gives us

$\frac{\alpha U}{P_1 X_1}=\frac{(1-\alpha) U}{P_2 X_2}$

which allows us to solve for $X_1$ in terms of $X_2$ such that:

$X_1=X_2\frac{alpha P_2}{(1-\alpha) P_1 }$

which we now substitute into the budget constraint, such that:

$M = P_1X_2\frac{\alpha P_2}{(1-\alpha) P_1 } + P_2 X_2$

$(1-\alpha) M =  \alpha P_2 X_2 + (1-\alpha) P_2 X_2$

$(1-\alpha) M =  \alpha P_2 X_2 +  P_2 X_2 - \alpha P_2 X_2$

$(1-\alpha) M =  \alpha P_2 X_2 +  P_2 X_2 - \alpha P_2 X_2$

$(1-\alpha)M = P_2 X_2$

Which can be re-arranged to give us the demand equation, which we denote as $X_2^*$

$X_2^* = \frac{(1-\alpha)M}{P_2}$

By symmetry, we should hopefully see that:

$X_1^* = \frac{(\alpha)M}{P_1}$

Substituting both these demand equations into our utility function gives the indirect utility function:

$V(P_1,P_2,M) = U(X_1^*,X_2^*) = (\frac{(\alpha)M}{P_1} )^\alpha (\frac{(\alpha)M}{P_2})^{1-\alpha}$

$V(P_1,P_2,M) = M(\frac{(\alpha)}{P_1})^\alpha (\frac{(\alpha)}{P_2})^{1-\alpha}$