Indirect Utility Function Calculator

Formula first

Overview

The Indirect Utility Function, denoted as $v(\mathbf{p},m)$, represents the highest level of utility an individual can attain given a set of prices for goods ($\mathbf{p}$) and their total income ($m$). It is derived by solving the consumer's utility maximization problem, where the consumer chooses a consumption bundle ($\mathbf{x}$) to maximize their direct utility function ($U(\mathbf{x})$) subject to a budget constraint ($\mathbf{p}\cdot \mathbf{x}\le m$). This function is crucial for analyzing how changes in prices and income affect a consumer's well-being.

Symbols

Variables

$\mathbf{p}$ = Price Vector, m = Income, v = Indirect Utility

Apply it well

When To Use

When to use: This equation is used when you need to determine the maximum utility a consumer can achieve given specific market prices and their budget. It's particularly useful for welfare analysis, comparing consumer well-being across different economic conditions, or evaluating the impact of policy changes (e.g., taxes or subsidies) on purchasing power.

Why it matters: The Indirect Utility Function is fundamental in microeconomics for understanding consumer behavior and welfare. It provides a direct link between market conditions (prices and income) and a consumer's utility, allowing economists to analyze demand theory, derive compensated demand functions, and assess the real income effects of price changes.

Avoid these traps

Common Mistakes

• Confusing the Indirect Utility Function with the Direct Utility Function $U(\mathbf{x})$.
• Attempting to include the consumption bundle $\mathbf{x}$ as an argument of $v(\mathbf{p},m)$.
• Incorrectly solving the underlying utility maximization problem, leading to an incorrect $v(\mathbf{p},m)$.

One free problem

Practice Problem

A consumer has a utility function $U(x_1,x_2)=x_1{x}_2$. The prices of goods are $p_1=2$ and $p_2=4$, and the consumer's income is $m=100$. Calculate the Indirect Utility Function value for this consumer.

Solve for: $v$

Hint: First find the Marshallian demand functions for $x_1$ and $x_2$, then substitute them into the utility function.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Microeconomic Theory by Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green
2. Microeconomics by Hal R. Varian
3. Wikipedia: Indirect utility function
4. Varian, Hal R. Microeconomic Analysis. 3rd ed. W. W. Norton & Company, 1992.
5. Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. Oxford University Press, 1995.
6. Hal R. Varian, Microeconomic Analysis
7. Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory
8. Varian, Hal R. Microeconomic Analysis. W. W. Norton & Company, 3rd ed., 1992, Chapter 7.