<p>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 $y$. </p>

<p>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 $p\cdot x \le w$.</p>

<p>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 $w'$.</p>

CC BY-SA 3.0
2016-04-03T18:35:38.893Z
The indirect utility function also gives information about the optimal consumption bundle for a given price-wealth pair. If the IUF is differentiable, you can get the optimal bundle by taking derivatives: https://en.m.wikipedia.org/wiki/Roy%27s_identity
manofbear
CC BY-SA 3.0
2016-04-03T18:28:07.473Z
2016-04-03T18:28:07.473Z
manofbear
<p>This is my opinion.
If indirect utility function is important, that is because of Duality theorem, envelope theorem.</p>

<p>Indirect utility function is one of a measure for consumer welfare. This is an optimized value from UMP. UMP is based on ordinal utility function, so indirect utility function can't be used for comparing consumer welfare.
So if you want to compare consumer welfare, you should use expenditure function(which is based on CMP).</p>

CC BY-SA 3.0
2016-04-03T14:32:14.640Z
2016-04-03T14:32:14.640Z
user7845
<p>It is very useful in industrial organization, whereby extensive use of discrete choice econometric models is made. See for instance McFadden 1973. </p>

CC BY-SA 3.0
2016-04-09T17:40:51.170Z
2016-04-09T17:40:51.170Z
rsm
<p>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.</p>

<microeconomics><utility><consumer-theory><choice-theory>
What is the point of the indirect utility function?
TheEconomist
# What is the point of the indirect utility function?

(Asked by: username_0 on 2016-04-03)

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.

**Tags:** microeconomics, utility, consumer-theory, choice-theory

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## 3 answers

### Answer by username_1 on 2016-04-03. Score: 5

Recall that if $x$ and $y$ are consumption bundles, $u(x)$ is the consumer's utility function, and $u(x)>u(y)$ 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: $v(p,w)>v(p',w')$ 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 $w'$.

> **Comments:**
> **username_1**: The indirect utility function also gives information about the optimal consumption bundle for a given price-wealth pair. If the IUF is differentiable, you can get the optimal bundle by taking derivatives: https://en.m.wikipedia.org/wiki/Roy%27s_identity  

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### Answer by username_2 on 2016-04-03. Score: 1

This is my opinion.
If indirect utility function is important, that is because of Duality theorem, envelope theorem.

Indirect utility function is one of a measure for consumer welfare. This is an optimized value from UMP. UMP is based on ordinal utility function, so indirect utility function can't be used for comparing consumer welfare.
So if you want to compare consumer welfare, you should use expenditure function(which is based on CMP).


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### Answer by username_3 on 2016-04-09. Score: 0

It is very useful in industrial organization, whereby extensive use of discrete choice econometric models is made. See for instance McFadden 1973.


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