What is the difference between direct and indirect utility? Following the advanced microeconomics textbook Jehle and Reny I can not understand either the intuition or the math definition. It confuses me.
Something that may help with the intuition:
U <- Goods <- Income Prices
The direct utility is derived from the consumption of goods. Said it simply, money, on its own, is almost worthless in terms of utility (setting aside cases where people simply like having money). Goods, on the other hand, generate utility because they can be used or consumed directly.
Now, Goods can be obtained only with money, and based on their prices (how much you can buy). So, indirectly (notice the framework i started with), income does not generate utility, but it does generate utility "indirectly", because it allows us to buy goods.
The main difference is in the domain of the functions
A utility function for a consumer is a function $u(x)$ of the consumption bundle that represents their preferences over consumption bundles.
Given a utility function $u$, the indirect utility function for a consumer is a function $v(\omega,p)$ of their wealth and the market prices that indicates the maximum utility they can achieve taking their wealth and the market prices as given.
I am using the notation $u$ of utility, $v$ for indirect utility, $p$ for prices, and $\omega$ for wealth. (I don't have a copy of the book with me. I don't know their notation.)