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| − | Analogously, in Producer Theory, the [[Conditional factor demand]] function can be related to the cost function by |
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| − | <math>x_{i}({\mathbf{w}},y) = \partial c(\mathbf{p}, y)/\partial p_{i} \,</math> |
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== Derivation == |
== Derivation == |
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| − | 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. |
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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: |
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| − | + | <math>\mathcal{L}(\mathbf{x}) = px-\lambda (U(x)-u)</math> with the FOCs: |
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| − | + | <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> |
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| − | + | <math>\partial \mathcal{L}/ \partial \lambda = 0 \leftrightarrow U(x) = u</math> |
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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> |
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| − | 2) Differentiate the optimal expenditure function with respect to <math>p_{i} \, |
+ | 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> |
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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: |
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| − | + | <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> |
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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>: |
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| − | + | <math> \partial U(x)/\partial x_{i} = 0\,</math> |
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5) Inserting this optimality condition into (3), we get the result: |
5) Inserting this optimality condition into (3), we get the result: |
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| − | + | <math> h_{i}(\mathbf{p},u) =\partial e(p,u)/ \partial p_{i} </math> |
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| − | [[Category:Microeconomics]] |
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