<p>Fix a utility level <span class="math-container">$u$</span>. For all prices <span class="math-container">$p$</span> we have <span class="math-container">$u(h(p,u))=u$</span>, so the function <span class="math-container">$u(h(p,u))$</span> is constant in prices. Thus, its derivative w.r.t. <span class="math-container">$p$</span> is <span class="math-container">$0$</span>. Applying the chain rule gives you <span class="math-container">$\nabla u(h(p,u)) ~\cdot~D_p h(p,u)=0$</span>.</p>

CC BY-SA 4.0
2023-02-01T09:07:58.933Z
2023-02-01T09:07:58.933Z
VARulle
<p>Mathematically speaking, the reason is streightforward. If <span class="math-container">$u(h(p,u))=u$</span> for any price level, if you take its derivative w.r.t. <span class="math-container">$p$</span>, you get zero. Note that you must be convinced that the utility function evaluated at <span class="math-container">$h(p,u)$</span> is equal to <span class="math-container">$u$</span>, because  <span class="math-container">$h(p,u)$</span> is the solution of the expenditure minimization problem, and <span class="math-container">$u$</span> is the utility target satisfied, by definition, by <span class="math-container">$h(p,u)$</span> because <span class="math-container">$h(p,u)$</span> 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 <span class="math-container">$h(p,u)$</span> is a solution of your expenditure minimization problem.</p>
<p>The economic interpretation of <span class="math-container">$\nabla_p e(p,u)=h(p,u)$</span> 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, <span class="math-container">$h(p,u)$</span>), and you can disregard the indirect effect on expenditure caused by the induced change in demand holding prices fixed, <span class="math-container">$p \cdot D_p h(p,u)$</span></p>

CC BY-SA 4.0
2023-02-01T11:04:30.460Z
2023-02-01T11:04:30.460Z
Tony
<p>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 <span class="math-container">$l$</span>, with <span class="math-container">$l=1,2,...,L$</span>, yields the hicksian demand function.</p>
<p>My reasoning is:</p>
<p><span class="math-container">$\nabla_p e(p,u) = \nabla_p \left[p ~\cdot~h(p,u)\right]$</span>,</p>
<p>where, <span class="math-container">$e(p,u)$</span> is the expenditure function, which is the value function of the expenditure minimization problem (EMP). By definition, <span class="math-container">$e(p,u) = p \cdot h(p,u)$</span>, where <span class="math-container">$h(p,u)$</span> is the hicksian demand correspondence, that is, the set of consumption bundles solving the EMP at a price vector <span class="math-container">$p$</span>.</p>
<p>I just apply the chain rule, and I write:</p>
<p><span class="math-container">$\nabla_p e(p.u)=h(p,u) + [p ~\cdot~D_p h(p,u)]^{\top}$</span></p>
<p>where <span class="math-container">$D_p h(p,u)$</span> 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), <span class="math-container">$p= \lambda \nabla u(h(p,u))$</span>, yields</p>
<p><span class="math-container">$\nabla_p e(p.u)=h(p,u) + \lambda[ \nabla u(h(p,u)) ~\cdot~D_p h(p,u)]^{\top}$</span></p>
<p>Then the book says: since the constraint <span class="math-container">$u(h(p,u))=u$</span> holds for all <span class="math-container">$p$</span> in the EMP, we know that <span class="math-container">$\nabla u(h(p,u)) ~\cdot~D_p h(p,u)=0$</span>, and so we have <span class="math-container">$h(p,u)=\nabla e(p,u)$</span>.</p>
<p>Why <span class="math-container">$\nabla u(h(p,u)) ~\cdot~D_p h(p,u)=0$</span> ? It is about the fact that <span class="math-container">$\nabla u(h(p,u))=0$</span> because the Hicksian demand correspondence <span class="math-container">$h(p,u)$</span> statisfies the property of no-excess utility?</p>

<microeconomics><consumer-theory>
Equivalent of shephard's lemma in consumer theory
John M.
# Equivalent of shephard's lemma in consumer theory

(Asked by: username_0 on 2023-01-31)

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?

**Tags:** microeconomics, consumer-theory


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## 2 answers

### Answer by username_2 on 2023-02-01. Score: 2 | Accepted answer

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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### Answer by username_1 on 2023-02-01. Score: 2

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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