FANDOM


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The Hicksian demand function can be related to the expenditure function by
 
The Hicksian demand function can be related to the expenditure function by
   
<math>h_{i}({\mathbf{p}}, u) = \partial e(\mathbf{p}, u)/\partial p_{i} \,</math>
+
<math>h_{i}({\mathbf{p}}, u) = \partial e(\mathbf{p}, u)/\partial p_{i} \,</math>
   
 
== Derivation ==
 
== Derivation ==
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1) The expenditure function returns the cost-minimizing value for reaching a fixed level of utility <math>u \,</math> at given prices <math>\mathbf{p}</math>. As such, it is the result of a constrained optimization problem:
 
1) The expenditure function returns the cost-minimizing value for reaching a fixed level of utility <math>u \,</math> at given prices <math>\mathbf{p}</math>. As such, it is the result of a constrained optimization problem:
   
<math>\mathcal{L}(\mathbf{x}) = px-\lambda (U(x)-u)</math> with the FOCs:
+
<math>\mathcal{L}(\mathbf{x}) = px-\lambda (U(x)-u)</math> with the FOCs:
   
<math>\partial \mathcal{L}/\partial{x_{i}} = 0 \leftrightarrow p_{i}=\lambda \partial U(x)/ \partial x_{i} </math> for <math> i=1,...k \,</math>
+
<math>\partial \mathcal{L}/\partial{x_{i}} = 0 \leftrightarrow p_{i}=\lambda \partial U(x)/ \partial x_{i} </math> for <math> i=1,...k \,</math>
   
<math>\partial \mathcal{L}/ \partial \lambda = 0 \leftrightarrow U(x) = u</math>
+
<math>\partial \mathcal{L}/ \partial \lambda = 0 \leftrightarrow U(x) = u</math>
   
where the resulting optimal <math>\mathbf{x\,}</math> are given by <math>x_{i}^{*}=h_{i}(p_{i},u) \,</math> and the cost-minimizing expenditure is given by <math>e(p,U)=h(p,u) \cdot p \,</math>
+
where the resulting optimal <math>\mathbf{x\,}</math> are given by <math>x_{i}^{*}=h_{i}(p_{i},u) \,</math> and the cost-minimizing expenditure is given by <math>e(p,U)=h(p,u) \cdot p \,</math>
   
2) Differentiate the optimal expenditure function with respect to <math>p_{i} \,</math> using the Chain rule:
+
2) Differentiate the optimal expenditure function with respect to <math>p_{i} \, </math> using the Chain rule:
   
<math>\partial e(p,u) / \partial p_{i} = \partial h_{i}(\mathbf{p},u)\cdot p_{i} + h_{i}(\mathbf{p}, u)</math>
+
<math>\partial e(p,u) / \partial p_{i} = \partial h_{i}(\mathbf{p},u)\cdot p_{i} + h_{i}(\mathbf{p}, u)</math>
   
 
3) Replace <math>p_{i}\,</math> by the FOC of the original optimization problem:
 
3) Replace <math>p_{i}\,</math> by the FOC of the original optimization problem:
   
<math>\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)</math>
+
<math>\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)</math>
   
 
4) Differentiate the budget constraint FOC from (1) with respect to <math>x_{i}\,</math>:
 
4) Differentiate the budget constraint FOC from (1) with respect to <math>x_{i}\,</math>:
   
<math> \partial U(x)/\partial x_{i} = 0\,</math>
+
<math> \partial U(x)/\partial x_{i} = 0\,</math>
   
 
5) Inserting this optimality condition into (3), we get the result:
 
5) Inserting this optimality condition into (3), we get the result:
   
<math> h_{i}(\mathbf{p},u) =\partial e(p,u)/ \partial p_{i} </math>
+
<math> h_{i}(\mathbf{p},u) =\partial e(p,u)/ \partial p_{i} </math>

Revision as of 00:39, November 11, 2011

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}   $
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