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In Consumer Theory, the Hicksian demand function can be related to the expenditure function by




Analogously, in Producer Theory, the Conditional factor demand function can be related to the cost function by




## Derivation

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.

1) The expenditure function returns the cost-minimizing value for reaching a fixed level of utility at given prices . As such, it is the result of a constrained optimization problem:

 with the FOCs:

 for




where the resulting optimal are given by and the cost-minimizing expenditure is given by

2) Differentiate the optimal expenditure function with respect to using the Chain rule:




3) Replace by the FOC of the original optimization problem:




4) Differentiate the budget constraint FOC from (1) with respect to :




5) Inserting this optimality condition into (3), we get the result:



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