Roy's identity

Roy's identity derives a consumer's ordinary demand for a good from the indirect utility function, as the negative ratio of its price derivative to its income derivative.

Indirect utility function: The maximum utility attainable at given prices and income. Roy's identity starts from this function and extracts demand from its derivatives.
Utility maximization: The choice of the most preferred feasible bundle under a specified constraint. The identity represents choices generated by maximizing utility subject to a budget.
Consumer demand theory: The study of how consumers allocate income among goods at given prices. Roy's identity is a central bridge from utility representations to demand predictions.
Hicksian demand: The least-cost bundle that attains a specified utility level at given prices. Unlike Roy's ordinary demand, it holds utility fixed while prices change.
Shephard's lemma: A result that derives compensated demand from derivatives of the expenditure function. It uses a parallel derivative method, but yields compensated rather than ordinary demand.
Budget constraint: A limit on affordable consumption bundles determined by prices and income. It is the constraint whose relaxation and price changes appear in the identity.
Demand estimation: The statistical inference of relationships between quantities demanded, prices, and income. A specified indirect utility function can imply a parametric demand system through Roy's identity.
Expenditure function: The minimum expenditure needed to attain a given utility level at specified prices. Its derivatives yield Hicksian demand, the dual counterpart to Roy's identity.
Marshallian demand: The quantities of goods chosen to maximize utility at given prices and income. This is the ordinary demand Roy's identity recovers.
Differentiability: A property of a function that permits a well-defined derivative at a point or throughout a region. The ratio requires the indirect utility function to have usable price and income derivatives.

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