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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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The following derivation is for relationship between the Hicksian demand and the expenditure function. The relationship between conditional factor demand and the cost function is analog. 





1) The expenditure function returns the costminimizing 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 costminimizing 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: 