Compensated (Hicksian) Demand Function Calculator

Formula first

Overview

The Compensated (Hicksian) Demand Function, derived from Shephard's Lemma, describes the quantity of a good a consumer would demand to achieve a specific utility level, assuming their income is 'compensated' for price changes. Unlike Marshallian demand, Hicksian demand isolates the substitution effect by holding utility constant, making it a crucial concept in welfare economics for analyzing the true cost of living and the impact of price changes on consumer well-being, free from income effects.

Symbols

Variables

$\mathbf{p}$ = Price Vector, u = Utility Level, e = Expenditure Function, $p_i$ = Price of Good i, $h_i$ = Hicksian Demand for Good i

Apply it well

When To Use

When to use: This formula is used in microeconomics to derive the Hicksian demand function for a good when the expenditure function is known. It's essential for analyzing consumer behavior under the assumption of constant utility, particularly when separating substitution effects from income effects of price changes, or for welfare analysis.

Why it matters: Understanding Hicksian demand is fundamental for advanced consumer theory and welfare economics. It allows economists to precisely measure the welfare impact of price changes (e.g., using compensating variation or equivalent variation) and to construct true cost-of-living indices, providing a more accurate picture of consumer well-being than standard Marshallian demand.

Avoid these traps

Common Mistakes

• Confusing Hicksian demand with Marshallian demand (which holds income constant).
• Incorrectly performing the partial differentiation, especially with multiple price variables.
• Forgetting that $\mathbf{p}$ is a vector of *all* prices, not just $p_i$.

One free problem

Practice Problem

Given an expenditure function $e(\mathbf{p},u)=u\sqrt{p_1{p}_2}$, where $p_1$ and $p_2$ are prices of two goods and $u$ is the utility level. Derive the Hicksian demand function for good 1, $h_1(\mathbf{p},u)$.

Solve for: $h1$

Hint: Apply the partial derivative rule: $\frac{\partial }{\partial x}(a{x}^n)=an{x}^{n-1}$ and chain rule if needed.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Varian, Hal R. Microeconomic Analysis. W. W. Norton & Company.
2. Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. Oxford University Press.
3. Wikipedia: Hicksian demand function
4. Wikipedia: Shephard's lemma
5. Microeconomic Analysis, 3rd Edition by Hal R. Varian
6. Microeconomic Theory: Basic Principles and Extensions, 12th Edition by Walter Nicholson and Christopher Snyder
7. Nicholson, Walter, and Christopher Snyder. Microeconomic Theory: Basic Principles and Extensions. Cengage Learning.
8. Shephard, R. W. (1953). Cost and Production Functions. Princeton University Press. (Formal proof of Shephard's Lemma)