Shephard's lemma

Shephard's lemma states that the derivative of a cost function with respect to an input price equals the cost-minimizing demand for that input, under suitable regularity conditions.

Cost function: A function giving the minimum expenditure needed to produce specified output at given input prices. Shephard's lemma differentiates this function to recover the minimizing input choices.
Production function: A function describing the maximum output obtainable from specified quantities of inputs. It defines the feasible production choices over which the firm minimizes cost.
Cost-function estimation: The econometric estimation of a firm's minimum expenditure as a function of output and input prices. Estimated cost derivatives can be translated into implied input demands.
Roy's identity: A result deriving Marshallian demand from derivatives of the indirect utility function. It recovers consumer demand from an indirect value function rather than firm cost.
Conditional factor demand: The cost-minimizing quantity of each production input at given prices and output. It is the input-demand quantity obtained by differentiating cost with respect to its price.
Cost minimization: The problem of producing a target output at the lowest possible expenditure. The lemma rests on how its optimal expenditure responds to input prices.
Translog cost function: A flexible second-order approximation to a logarithmic cost function in prices and output. Its price derivatives generate practical estimates of conditional input demands.
Marshallian demand: Consumer demand that maximizes utility subject to prices and an income constraint. Unlike conditional input demand, it varies with income rather than holding output fixed.
Hotelling's lemma: A duality result stating that derivatives of a firm's profit function yield its net supply choices. It is the profit-maximization counterpart to Shephard's cost-minimization result.
Input price: The monetary cost per unit of a resource used in production. It is the variable with respect to which the cost function is differentiated.

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