What is the point of the indirect utility function? Where does this have application? I understand how the demand function may be arrived at using the utility maximization problem but I don't understand where the indirect utility function is used and I think the two are somewhat related... Please excuse me if this is a very fundamental question. I have only started Intermediate Microeconomics.
Recall that if $x$ and $y$ are consumption bundles, $u(x)$ is the consumer's utility function, and $u(x)&gt;u(y)$ means the consumer strictly prefers bundle $x$ to bundle $y$. 

The indirect utility function $v(p,w)$ is the highest value of the utility function when $u$ is evaluated over all affordable bundles given $(p,w)$. In other words, $v(p,w)=max u(x)$ subject to $p\cdot x \le w$.

So the indirect utility function tells you the consumers preferences over price-wealth pairs: $v(p,w)&gt;v(p',w')$ means the consumer gets to consume a strictly better bundle when prices are $p$ and wealth is $w$ than any other affordable bundle when prices and wealth are $p'$ and $w'$.