Misologia …

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Shephard's Lemma

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<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>
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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>
   
 
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:
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