## FANDOM

15 Pages

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

This derivation uses the envelope theorem.

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:

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

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