Shephard's Lemma

5) The Hicksian demand function can be related to the expenditure function by

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

Derivation

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:

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