<p>I got a homework exercise that I am stuck with: Use Shephard's lemma to show that price elasticities of Hicksian Demand <span class="math-container">$\varepsilon_{jk}^{h}=\frac{\partial x_{j}^{h}}{\partial p_{k}} \frac{p_{k}}{x_{j}^{h}}$</span> satisfy the equation: <span class="math-container">$\varepsilon_{jj}^{h}\varepsilon_{kk}^{h}\geq \varepsilon_{jk}^{h}\varepsilon_{kj}^{h}$</span> .</p>
<p>I've spent nearly 3 hours of honest work (the result is not impressive, but it was honest work) and found the following:</p>
<p>The Shephard's Lemma: <span class="math-container">$x^{_{k}^{h}}(\mathbf{p},U)=\frac{\partial e(\mathbf{p},U])}{\partial p_{k}}$</span></p>
<p>Cross-price elasticity: <span class="math-container">$\epsilon_{kj} = \frac{\partial x_k}{\partial p_j}\frac{p_j}{x_k}$</span></p>
<p>The symmetry of Hicksian Demand (Jehle &amp; Reny p. 62): <span class="math-container">$\frac{\partial x_{k}^{h}(\mathbf{p},y)}{\partial p_j}=\frac{\partial x_{j}^{h}(\mathbf{p},y)}{\partial p_k}$</span> .</p>
<p>This is all I could find on this topic. Maybe if not the for whole solution I could ask for a hint on how to take the next step. I understand that to use Shephard's Lemma I should somehow obtain the expenditure function e(p, U). But how to do it is beyond me.</p>
<p>My interpretation of the required answer is to show that the own-price elasticity of goods <em>i</em> and <em>j</em> i greater than their cross-price elasticity (<em>ij</em> which is symmetrical to <em>ji</em>).</p>

CC BY-SA 4.0
2022-12-03T10:41:36.330Z
Hi, we have rule here that homework/self-study questions are only allowed if author shows their honest attempt at trying solve the problem himself/herself. Please add your attempt to the question
1muflon1
CC BY-SA 4.0
2022-12-03T12:10:54.580Z
@1muflon1 Thank you - I didn't know that. I will do my best and then post it here.
Barty82
CC BY-SA 4.0
2022-12-05T21:56:40.467Z
You could use the fact that a Hicksian demand system has a nsd Hessian matrix...
Bertrand
<microeconomics>
Show that Hicksian Demand price elasticities satisfy a given equation (Use Shephard's Lemma)
Barty82
# Show that Hicksian Demand price elasticities satisfy a given equation (Use Shephard's Lemma)

(Asked by: username_0 on 2022-12-03)

I got a homework exercise that I am stuck with: Use Shephard's lemma to show that price elasticities of Hicksian Demand $\varepsilon\_{jk}^{h}=\frac{\partial x\_{j}^{h}}{\partial p\_{k}} \frac{p\_{k}}{x\_{j}^{h}}$ satisfy the equation: $\varepsilon\_{jj}^{h}\varepsilon\_{kk}^{h}\geq \varepsilon\_{jk}^{h}\varepsilon\_{kj}^{h}$ .

I've spent nearly 3 hours of honest work (the result is not impressive, but it was honest work) and found the following:

The Shephard's Lemma: $x^{\_{k}^{h}}(\mathbf{p},U)=\frac{\partial e(\mathbf{p},U])}{\partial p\_{k}}$

Cross-price elasticity: $\epsilon\_{kj} = \frac{\partial x\_k}{\partial p\_j}\frac{p\_j}{x\_k}$

The symmetry of Hicksian Demand (Jehle & Reny p. 62): $\frac{\partial x\_{k}^{h}(\mathbf{p},y)}{\partial p\_j}=\frac{\partial x\_{j}^{h}(\mathbf{p},y)}{\partial p\_k}$ .

This is all I could find on this topic. Maybe if not the for whole solution I could ask for a hint on how to take the next step. I understand that to use Shephard's Lemma I should somehow obtain the expenditure function e(p, U). But how to do it is beyond me.

My interpretation of the required answer is to show that the own-price elasticity of goods *i* and *j* i greater than their cross-price elasticity (*ij* which is symmetrical to *ji*).

**Tags:** microeconomics

> **Comments:**
> **username_1**: Hi, we have rule here that homework/self-study questions are only allowed if author shows their honest attempt at trying solve the problem himself/herself. Please add your attempt to the question  
> **username_0**: @username_1 Thank you - I didn't know that. I will do my best and then post it here.  
> **username_2**: You could use the fact that a Hicksian demand system has a nsd Hessian matrix...  

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## 0 answers

