Intuition of Shephard's Lemma in Reverse I am not sure how to word this question. I understand the envelop theorem pretty well and as a result Shepard's Lemma. The proof is relatively straightforward. I am getting stuck on the intuition behind integrating the Hicksian demand function and getting the (total) expenditure function as a result. It seems like this would be some sort of reversal of the envelope theorem.
Suppose we have a two good world. A typical Hicksian demand function can be written as a function of p1, p2, and utility. Using the fundamental theorem of calculus and the envelop theorem, I know that integrating this function over an interval with respect to p1 or p2 will give us back the total expenditure function. So if we integrate over a range with respect to p1, for example, it gives us the change in total expenditure. What's not clear to me is how expenditures on good 2 are being included (intuitively).
In a single-variable integral, it is easy to understand this intuitively. If we have x and we integrate a function f(x) over an interval it is easy to understand this as adding up the infinitely small changes times the rate of change of f'(x) to get the &quot;total&quot; change in f(x). In the Hicksian case, we seem to be adding up the infinitely small changes in price of good 1 times the rate of change (Hicksian demand for good 1) and assuming expenditures on good 2 are constant when they almost certainly wouldn't be. Is there a better way to understand this? Are expenditures on good 2 implicitly included?
Expenditures on good $2$ are, indeed, included and can vary.
We know from Shephard's lemma that whenever the marginal change in expenditure for good $1$
with respect to its price varies with the price of good $1$, the Hicksian demand for good $1$ must vary too. But since utility is fixed, changes in the Hicksian demand for good $1$ require changes in the Hicksian demand for good $2$, a move along the indifference curve. And since we do not change the price of good $2$, the expenditure on good $2$ must vary too.