What is Roy’s Identity: How to Link Indirect Utility to Marshallian Demand (microeconomics tutoring)

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Roy’s Identity: Linking Indirect Utility to Marshallian Demand (microeconomics tutoring)
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Roy’s Identity is a cornerstone of microeconomics tutoring and consumer theory. It provides a direct way to recover Marshallian demand from the indirect utility function, linking dual representations of preferences.

Students often struggle with the intuition behind the identity and the steps in its derivation. If you need help with consumer theory, duality, or exam preparation, visit the tutoring services page.

Roy’s Identity: Marshallian demand for good i = – (∂v/∂pᵢ) / (∂v/∂m), where v(p, m) is the indirect utility function.

The identity shows how optimal consumption choices can be extracted from the way utility responds to price changes and income changes. It is a powerful bridge between indirect utility and observable demand behavior.

Why Roy’s Identity Matters

Roy’s Identity is essential because it:

  • connects dual and primal consumer theory
  • allows demand to be derived without solving utility maximization directly
  • is foundational for welfare analysis and comparative statics
  • appears frequently in graduate micro exams

How to Derive Roy’s Identity Step by Step

  1. Start with the indirect utility function:
    v(p, m) = maxₓ u(x) subject to p·x ≤ m.
  2. Use the envelope theorem:
    ∂v/∂pᵢ = -λxᵢ* ∂v/∂m = λ where λ is the marginal utility of income.
  3. Take the ratio:
    (∂v/∂pᵢ) / (∂v/∂m) = -xᵢ*
  4. Rearrange:
    xᵢ*(p, m) = – (∂v/∂pᵢ) / (∂v/∂m)
  5. Interpretation:
    Demand is recovered from how utility changes with prices and income.

Numerical Example

Suppose a consumer has Cobb–Douglas utility: u(x₁, x₂) = x₁^0.5 x₂^0.5.

The indirect utility function is: v(p₁, p₂, m) = (m² / (4 p₁ p₂))^0.5 = m / (2 √(p₁ p₂)).

Compute derivatives:

  • ∂v/∂p₁ = – m / (4 p₁ √(p₁ p₂))
  • ∂v/∂m = 1 / (2 √(p₁ p₂))

Apply Roy’s Identity:

x₁* = – (∂v/∂p₁) / (∂v/∂m) = (m / (4 p₁ √(p₁ p₂))) × (2 √(p₁ p₂)) = m / (2 p₁)

This matches the known Marshallian demand for Cobb–Douglas.

Common Mistakes

  • Forgetting the negative sign in the formula
  • Confusing Marshallian and Hicksian demand
  • Misapplying the envelope theorem
  • Using total derivatives instead of partial derivatives

Why This Matters in Microeconomics

Roy’s Identity is central to:

  • consumer theory and duality
  • deriving demand without solving optimization problems
  • welfare analysis and policy evaluation
  • graduate-level microeconomic modeling

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