# Why is Roy's Identity so important?

I've been reviewing some of my microeconomics theory and have been reading up on Roy's Identity. Recall that Roy's identity is defined as: $$x^*_{i}(\text{p},m)=-\frac{\frac{\partial v}{\partial p_{i}}}{\frac{\partial v}{\partial m}}$$

being that the indirect utility function is just the utility function evaluated at the appropriate Marshallian demands, this result is not surprising.

What is an important application of roy's identity that I wouldn't know from just solving for marshallian demands?

It is not that surprising if you have the right intuition, but let's make sure we consider it unsurprising for the right reasons. Roy's identity can be rewritten as

$$x^*_{i}(\text{p},m)\frac{\partial v}{\partial m}=-\frac{\partial v}{\partial p_{i}}.$$ The right-hand side is the marginal utility loss from a price increase. The left-hand side is the amount of good $i$ consumed times the marginal utility of money. If the consumer were forced to consume exactly $x^*_{i}(\text{p},m)$ units of good $i$, this would be obvious.

The consumer has to pay proportionally more of the income on buying good $i$, and cannot spend it on other things whose marginal utility equals the marginal utility of money.

Of course, the consumer can make adjustments and is not forced to hold the amount fixed. But, similar as in Hotelling's lemma, the consumer who is at an optimum need not do any adjusting, indirect effects are negligible. And showing the latter is more than just evaluating utility at the appropriate Marshallian demands.

When working with assumed functional forms, then you could be justified supposing that there is not much added value.

However, when you try to model behavior without functional forms, you will need all the help you can get in terms of results like Roy's identity to prove your results.

One important application of Roy's identity is in deriving the Ramsey Rule for optimal commodity taxation. A derivation can be found here on page 14.