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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?

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2 Answers 2

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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, $p \cdot D_p h(p,u)$

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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 $\nabla u(h(p,u)) ~\cdot~D_p h(p,u)=0$.

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