Shephard’s Lemma: Hicksian Demand and the Expenditure Function

We can also estimate the Hicksian demands by using Shephard’s lemma which stats that the partial derivative of the expenditure function Ι with respect to the price i is equal to the Hicksian demand for good i. The general formula for Shephards lemma is given by

For good one we get:

We know since previously that the Hicksian demand for good one is given by Which can be written as This means in order for Shephards lemma to work the partial derivative of the expenditure function Ι with respect to the price of good one must be equal to this expression. Our expenditure function is given by If we substitute in the expressions for the Hicksian demand for good 1 and good 2 we get The partial derivative of the above function with respect to the price of good one is therefore given by

Which can be written as

Which we can see is the same expression that we had previously for the Hicksian demand for good one given by Which means that Shephards Lemma seams to work for good one.
For good two we get: We know since previously that the Hicksian demand for good two is given by Which can be written as

This means in order for Shephards lemma to work

the partial derivative of the expenditure function Ι with respect to the price of good two must be equal to this expression. Our expenditure function is given by If we substitute in the expressions for the Hicksian demand for good 1 and good 2 we get
The partial derivative of the above function with respect to the price of good two is therefore given by

Which can be written as

Which we can see is the same expression that we had previously for the Hicksian demand for good two given by. Which means that Shephards lemma seams to work for good two as well.