Definitions

# Shephard's lemma

Shephard's lemma is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost minimizing point of a given good ($i$) with price $p_i$ is unique. The idea is that a consumer will buy a unique ideal amount of each item to minimize the price for obtaining a certain level of utility given the price of goods in the market. It was named after Ronald Shephard who gave a proof using the distance formula in a paper published in 1953, although it was already used by John Hicks (1939) and Paul Samuelson (1947).

## Definition

The lemma give a precise formulation for the demand of each good in the market with respect to that level of utility and those prices: the derivative of the expenditure function ($e \left(p, u\right)$) with respect to that price:

$h_i\left(u, p\right) = frac\left\{partial e \left(p, u\right)\right\}\left\{ partial p_i\right\}$

where $h_i\left(u, p\right)$ is the Hicksian demand for good $i$, $e \left(p, u\right)$ is the expenditure function, and both functions are in terms of prices (a vector $p$) and utility $u$.

Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the envelope theorem.

## Application

Shephard's lemma gives a relationship between expenditure (or cost) functions and Hicksian demand. The lemma can be re-expressed as Roy's identity, which gives a relationship between an indirect utility function and a corresponding Marshallian demand function.