# Equivalent of shephard's lemma in consumer theory

I'm studying micro from the Mas-Colell, and I'm trying to understand the proof 2 of proposition 3.G.1. It is about proving that the derivative of the expenditure function w.r.t. the price of a commodity $$l$$, with $$l=1,2,...,L$$, yields the hicksian demand function.

My reasoning is:

$$\nabla_p e(p,u) = \nabla_p \left[p ~\cdot~h(p,u)\right]$$,

where, $$e(p,u)$$ is the expenditure function, which is the value function of the expenditure minimization problem (EMP). By definition, $$e(p,u) = p \cdot h(p,u)$$, where $$h(p,u)$$ is the hicksian demand correspondence, that is, the set of consumption bundles solving the EMP at a price vector $$p$$.

I just apply the chain rule, and I write:

$$\nabla_p e(p.u)=h(p,u) + [p ~\cdot~D_p h(p,u)]^{\top}$$

where $$D_p h(p,u)$$ is simply the derivative of the Hicksian w.r.t. prices. Then, substituting from the FOCs for an interior solution to the expenditure minimization problem (EMP), $$p= \lambda \nabla u(h(p,u))$$, yields

$$\nabla_p e(p.u)=h(p,u) + \lambda[ \nabla u(h(p,u)) ~\cdot~D_p h(p,u)]^{\top}$$

Then the book says: since the constraint $$u(h(p,u))=u$$ holds for all $$p$$ in the EMP, we know that $$\nabla u(h(p,u)) ~\cdot~D_p h(p,u)=0$$, and so we have $$h(p,u)=\nabla e(p,u)$$.

Why $$\nabla u(h(p,u)) ~\cdot~D_p h(p,u)=0$$ ? It is about the fact that $$\nabla u(h(p,u))=0$$ because the Hicksian demand correspondence $$h(p,u)$$ statisfies the property of no-excess utility?

Mathematically speaking, the reason is streightforward. If $$u(h(p,u))=u$$ for any price level, if you take its derivative w.r.t. $$p$$, you get zero. Note that you must be convinced that the utility function evaluated at $$h(p,u)$$ is equal to $$u$$, because $$h(p,u)$$ is the solution of the expenditure minimization problem, and $$u$$ is the utility target satisfied, by definition, by $$h(p,u)$$ because $$h(p,u)$$ is the set of optima. The property of no excess utility is peculiar for the hicksian demand corresponce because if violeted, it would contraddict that $$h(p,u)$$ is a solution of your expenditure minimization problem.
The economic interpretation of $$\nabla_p e(p,u)=h(p,u)$$ is: if you are at an optimum in the expenditure minimization problem, only the direct effect on the expenditure function matters (change in prices holding demand fixed, that is, $$h(p,u)$$), and you can disregard the indirect effect on expenditure caused by the induced change in demand holding prices fixed, $$p \cdot D_p h(p,u)$$
Fix a utility level $$u$$. For all prices $$p$$ we have $$u(h(p,u))=u$$, so the function $$u(h(p,u))$$ is constant in prices. Thus, its derivative w.r.t. $$p$$ is $$0$$. Applying the chain rule gives you $$\nabla u(h(p,u)) ~\cdot~D_p h(p,u)=0$$.