Shephard's Lemma

In Consumer Theory, the Hicksian demand function can be related to the expenditure function by

$$h_{i}({\mathbf{p}}, u) = \partial e(\mathbf{p}, u)/\partial p_{i} \,$$

Analogously, in Producer Theory, the Conditional factor demand function can be related to the cost function by $$x_{i}({\mathbf{w}},y) = \partial c(\mathbf{p}, y)/\partial p_{i} \,$$

Derivation
The following derivation is for relationship between the Hicksian demand and the expenditure function. The derivation for conditional factor demand and the cost function is identical, only with other notation.

1) The expenditure function returns the cost-minimizing value for reaching a fixed level of utility $$u \,$$ at given prices $$\mathbf{p}$$. As such, it is the result of a constrained optimization problem:

$$\mathcal{L}(\mathbf{x}) = px-\lambda (U(x)-u)$$ with the FOCs:

$$\partial \mathcal{L}/\partial{x_{i}} = 0 \leftrightarrow p_{i}=\lambda \partial U(x)/ \partial x_{i} $$ for $$ i=1,...k \,$$

$$\partial \mathcal{L}/ \partial \lambda = 0 \leftrightarrow U(x) = u$$

where the resulting optimal $$\mathbf{x\,}$$ are given by $$x_{i}^{*}=h_{i}(p_{i},u) \,$$ and the cost-minimizing expenditure is given by $$e(p,U)=h(p,u) \cdot p \,$$

2) Differentiate the optimal expenditure function with respect to $$p_{i} \, $$ using the Chain rule:

$$\partial e(p,u) / \partial p_{i} = \partial h_{i}(\mathbf{p},u)\cdot p_{i} + h_{i}(\mathbf{p}, u)$$

3) Replace $$p_{i}\,$$ by the FOC of the original optimization problem:

$$\partial e(\mathbf{p},u) / \partial p_{i} = \partial h_{i}(\mathbf{p},u) \cdot \lambda \partial U(x)/ \partial x_{i} + h_{i}(\mathbf{p},u)$$

4) Differentiate the budget constraint FOC from (1) with respect to $$x_{i}\,$$:

$$ \partial U(x)/\partial x_{i} = 0\,$$

5) Inserting this optimality condition into (3), we get the result:

$$ h_{i}(\mathbf{p},u) =\partial e(p,u)/ \partial p_{i}  $$