<p>Here's my attempt at a hint, though I appreciate feedback, as I am not sure how rigorous this is. As per the comment I gave with the resource, your question seems to require a previous question in order to do this.</p>

<p>So start by showing that the share of income spent on good $x_i$ can always be measured by the elasticity of expenditure function with respect to $p_i$ at $(p, u^*)$ where $u^* = v(p,w)$</p>

<p>$$\mid \eta^ {e_{x_i, p_i}}\mid \ = \frac{\frac{\Delta x_i}{x_i}}{\frac{\Delta p_i}{p_i}} = \frac{\Delta x_i}{\Delta p_i} \cdot \frac{p_i}{x_i} = \frac{\partial x_i}{\partial p_i} \cdot \frac{p_i}{x_i}$$</p>

<p>We know that,</p>

<p>$$e(p,u^*) = p \cdot \tilde x, \forall \ \tilde x ∈ x^h (p, u^*) = x(p, e(p, u^*))$$</p>

<p>by duality, where $x^h(p, u^*)$ denotes Hicksian demand. So,</p>

<p>$$\frac{\partial e(p_i,u^*)}{\partial p_i} = x_i$$
$$\frac{\partial x_i}{\partial e(p_i, u^*)} = \frac{x_i}{w}$$</p>

<p>$$\implies \frac{\partial x_i}{\partial p_i} \cdot \frac{p_i}{x_i} = \frac{\partial e}{\partial p_i} \cdot \frac{\partial x_i}{\partial e}\cdot \frac{p_i}{x_i} = \frac {x_i^2}{w} \cdot \frac{p_i}{x_i} = \frac{p_i x_i}{w}$$</p>

<p>Which is the share of income spent on good $x_i$</p>

CC BY-SA 3.0
2015-10-18T19:38:01.480Z
I have no clue why you post an answer after finding the question online.
Giskard
CC BY-SA 3.0
2015-10-18T19:56:07.140Z
I found the question online, but not the answer. This doesn't even answer his question; it answers a pre-cursor question. If the user is really trying then they will hopefully be able to use this. Are you just generally wary of giving any sort of help to those who ask homework questions? If you really think it's inappropriate, I can remove it, but there's no need to be so terse.
Kitsune Cavalry
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2015-10-18T20:59:35.260Z
Sorry, I have just now realized that what you gave was indeed a hint, not the full answer. It seemed very elaborate at first. I think the general policy is that we may help (give hints as you did) as long as it is clearly acknowledged that this is a homework question and the OP explains why/where/how he is stuck. Others may have differing opinions.
Giskard
CC BY-SA 3.0
2015-10-18T21:11:58.093Z
Fair enough. I'll try to make stronger nudges for people to show what they need help on.
Kitsune Cavalry
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2015-10-19T17:10:52.710Z
Sorry for harshing the stack exchange policy. I was in hurry; and I couldn't extend this question with my results.
Übel Yildmar
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2015-10-18T16:09:05.233Z
2015-10-18T16:09:05.233Z
Kitsune Cavalry
<p>Suppose that the expenditure function is multiplicatively separable in $\textbf{p}$ and u so that 
$$e(p,u) =k(u)g(\textbf{p}),$$
 where k(·) is some positive monotonic function of a single variable, and g: $\mathbb{R^n_{+}}$ $\rightarrow$ $\mathbb{R}_{+}$.</p>

<p>Thank you in advance.</p>

CC BY-SA 3.0
2015-10-18T11:21:55.990Z
After reading the title I tought I understood the question but then I read the body of it. Now I am not sure what you are asking. There seems to be some dissonance between the two? (Marshal vs. Hicks, income elasticiy of demand vs. price elasticity of income.) Could you please clarify? Either way this seems to be a fairly simple grad level exercise, so please show your own efforts at solution as well.
Giskard
CC BY-SA 3.0
2015-10-18T14:36:09.070Z
@denesp I found the question online, [number 6 in this set](http://homepage.univie.ac.at/ana-begona.ania-martinez/micro/micro_s2.pdf) I am still unsure what it is asking.
Kitsune Cavalry
CC BY-SA 3.0
2015-10-18T15:11:10.287Z
I guess from converting from Marshallian to Hicksian demand, we have to use a duality property between them.
Kitsune Cavalry
<microeconomics><elasticity>
How to show that the income elasticity of Marshallian demand is equal to unity?
Übel Yildmar
# How to show that the income elasticity of Marshallian demand is equal to unity?

(Asked by: username_0 on 2015-10-18)

Suppose that the expenditure function is multiplicatively separable in $\textbf{p}$ and u so that
$$e(p,u) =k(u)g(\textbf{p}),$$
where k(·) is some positive monotonic function of a single variable, and g: $\mathbb{R^n\_{+}}$ $\rightarrow$ $\mathbb{R}\_{+}$.

Thank you in advance.

**Tags:** microeconomics, elasticity
> **Comments:**
> **username_1**: After reading the title I tought I understood the question but then I read the body of it. Now I am not sure what you are asking. There seems to be some dissonance between the two? (Marshal vs. Hicks, income elasticiy of demand vs. price elasticity of income.) Could you please clarify? Either way this seems to be a fairly simple grad level exercise, so please show your own efforts at solution as well.  
> **username_2**: @denesp I found the question online, [number 6 in this set](http://homepage.univie.ac.at/ana-begona.ania-martinez/micro/micro_s2.pdf) I am still unsure what it is asking.  
> **username_2**: I guess from converting from Marshallian to Hicksian demand, we have to use a duality property between them.  

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## 1 answers

### Answer by username_2 on 2015-10-18. Score: 1

Here's my attempt at a hint, though I appreciate feedback, as I am not sure how rigorous this is. As per the comment I gave with the resource, your question seems to require a previous question in order to do this.

So start by showing that the share of income spent on good $x\_i$ can always be measured by the elasticity of expenditure function with respect to $p\_i$ at $(p, u^\*)$ where $u^\* = v(p,w)$

$$\mid \eta^ {e\_{x\_i, p\_i}}\mid \ = \frac{\frac{\Delta x\_i}{x\_i}}{\frac{\Delta p\_i}{p\_i}} = \frac{\Delta x\_i}{\Delta p\_i} \cdot \frac{p\_i}{x\_i} = \frac{\partial x\_i}{\partial p\_i} \cdot \frac{p\_i}{x\_i}$$

We know that,

$$e(p,u^\*) = p \cdot \tilde x, \forall \ \tilde x ∈ x^h (p, u^\*) = x(p, e(p, u^\*))$$

by duality, where $x^h(p, u^\*)$ denotes Hicksian demand. So,

$$\frac{\partial e(p\_i,u^\*)}{\partial p\_i} = x\_i$$
$$\frac{\partial x\_i}{\partial e(p\_i, u^\*)} = \frac{x\_i}{w}$$

$$\implies \frac{\partial x\_i}{\partial p\_i} \cdot \frac{p\_i}{x\_i} = \frac{\partial e}{\partial p\_i} \cdot \frac{\partial x\_i}{\partial e}\cdot \frac{p\_i}{x\_i} = \frac {x\_i^2}{w} \cdot \frac{p\_i}{x\_i} = \frac{p\_i x\_i}{w}$$

Which is the share of income spent on good $x\_i$

> **Comments:**
> **username_1**: I have no clue why you post an answer after finding the question online.  
> **username_2**: I found the question online, but not the answer. This doesn't even answer his question; it answers a pre-cursor question. If the user is really trying then they will hopefully be able to use this. Are you just generally wary of giving any sort of help to those who ask homework questions? If you really think it's inappropriate, I can remove it, but there's no need to be so terse.  
> **username_1**: Sorry, I have just now realized that what you gave was indeed a hint, not the full answer. It seemed very elaborate at first. I think the general policy is that we may help (give hints as you did) as long as it is clearly acknowledged that this is a homework question and the OP explains why/where/how he is stuck. Others may have differing opinions.  
> **username_2**: Fair enough. I'll try to make stronger nudges for people to show what they need help on.  
> **username_0**: Sorry for harshing the stack exchange policy. I was in hurry; and I couldn't extend this question with my results.  

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