<p>From (1) and (2) you get 
<span class="math-container">$$\frac{x_j}{x_i}=\frac{a_j p_i}{a_i p_j},$$</span>
or equivalently,
<span class="math-container">$$x_j =\frac{a_j p_i}{a_i p_j} x_i.$$</span>
Substituting this into equation 3 for <span class="math-container">$j=2,...,n$</span> and <span class="math-container">$i=1$</span> (solving for the demand function for good 1) we get
<span class="math-container">$$M=p_1x_1 + \sum_{j=2}^n p_j  \frac{a_j p_1}{a_1 p_j} x_1$$</span>
<span class="math-container">$$M=p_1x_1 + \sum_{j=2}^n  \frac{a_j p_1}{a_1} x_1$$</span>
<span class="math-container">$$M=p_1x_1 + \frac{p_1}{a_1} x_1\sum_{j=2}^n a_j $$</span>
<span class="math-container">$$M=p_1x_1(1 + \frac{1}{a_1}\sum_{j=2}^n a_j). $$</span>
Finally, solving for <span class="math-container">$x_1$</span> we get
<span class="math-container">$$x_1^* = \frac{M}{p_1}(1+ \frac{\sum_{j=2}^n a_j}{a_1})^{-1}$$</span>
<span class="math-container">$$x_1^* =\frac{a_1}{\sum_{j=1}^n a_j} \frac{M}{p_1}.$$</span></p>

<p>Analogously, 
<span class="math-container">$$x_i^*=\frac{a_i}{\sum_{j=1}^n a_j} \frac{M}{p_i},$$</span>
for <span class="math-container">$i=1,...,n$</span>.</p>

<p>As is always the case for demand functions derived from a Cobb-Douglas utility function, the consumer spends a constant share of income on each good. To see this, rearrange the previous equation to get
<span class="math-container">$$p_ix_i^*=\frac{a_i}{\sum_{j=1}^n a_j} M,$$</span>
for <span class="math-container">$i=1,...,n$</span>.</p>

CC BY-SA 4.0
2019-03-15T16:45:16.827Z
2019-03-15T17:08:00.607Z
2019-03-15T17:08:00.607Z
dlnB
<p>I derived demand, given a Cobb-Douglas utility function but I am not really sure if I did it correctly. I am especially struggling with the sum signs and the subscripts of <span class="math-container">$i$</span> &amp; <span class="math-container">$j$</span>. It would be really great if someone could check. I want to maximize utility for 2 goods, here <span class="math-container">$j$</span> and <span class="math-container">$i$</span>. </p>

<p><span class="math-container">$\ u(x_i)=\prod_{i=1}^n x^a_i       $</span></p>

<p><span class="math-container">$\ s.t.:M=\sum_{j=1}^n p_jx_j      $</span></p>

<p><span class="math-container">$\ L= \sum_{i=1}^n a_ilogx_i+\lambda(M-\sum_{j=1}^np_jx_j) $</span></p>

<p><span class="math-container">$ (1)\frac{\partial L}{\partial x_i} = \frac{a_i}{ x_i}-\lambda p_i=0$</span></p>

<p><span class="math-container">$ (2)\frac{\partial L}{\partial x_j} = \frac{a_j}{ x_j}-\lambda p_j=0$</span></p>

<p><span class="math-container">$ (3)\frac{\partial L}{\partial \lambda} = M-\sum_{j=1}^np_jx_j=0$</span></p>

<p>from (1) and (2) it follows:</p>

<p><span class="math-container">$ \frac{p_i}{x_i} = \frac{a_i/x_i}{a_j/x_j}=\frac{a_ix_j}{a_jx_i}$</span></p>

<p><span class="math-container">$ x_j = \frac{p_ja_jx_i}{p_ja_j}$</span></p>

<p><span class="math-container">$ x_j$</span> into (3)</p>

<p><span class="math-container">$ M = \sum_{j=1}^np_j(\frac{p_ja_jx_i}{p_j/a_j})=0$</span></p>

<p><span class="math-container">$ x_i=  \frac{a_iM}{\sum_{j=1}^na_jp_j}$</span></p>

<p>Furthermore I would like to interpret what happens if we have an efficency shock for good <span class="math-container">$i$</span>. Meaning, good <span class="math-container">$i$</span> becomes cheaper. This leads to an increase in relative income. So <span class="math-container">$M$</span> increases which leads to an increased demand for good <span class="math-container">$x_i$</span>, rest hold constant. Is that correct?</p>

CC BY-SA 4.0
2019-03-15T16:59:06.867Z
Sorry. Yes, a drop in $p_i$. And of course $M$ stays constant, but normally we say then that the relative income increases, since the household spends less money for each unit of $x_i$. So again, $p_i$ drops, this will increase the amount of $x_i$.
Pete
CC BY-SA 4.0
2019-03-15T17:05:40.483Z
His derivation is incorrect (due to a few algebra mistakes). The Cobb-Douglas utility function leads to a set of demand functions in which demand for each good depends only on it's own price. Check the two-good case for reference. See my solution below.
dlnB
CC BY-SA 4.0
2019-03-15T17:10:23.430Z
@dlnB: You're right. I misread $p_j$ in the denominator as $p_i$.
Herr K.
<utility><cobb-douglas><demand>
Demand derived from Cobb-Douglas utility, interpretation, check
Pete
# Demand derived from Cobb-Douglas utility, interpretation, check

(Asked by: username_0 on 2019-03-15)

I derived demand, given a Cobb-Douglas utility function but I am not really sure if I did it correctly. I am especially struggling with the sum signs and the subscripts of $i$ & $j$. It would be really great if someone could check. I want to maximize utility for 2 goods, here $j$ and $i$.

$\ u(x\_i)=\prod\_{i=1}^n x^a\_i $

$\ s.t.:M=\sum\_{j=1}^n p\_jx\_j $

$\ L= \sum\_{i=1}^n a\_ilogx\_i+\lambda(M-\sum\_{j=1}^np\_jx\_j) $

$ (1)\frac{\partial L}{\partial x\_i} = \frac{a\_i}{ x\_i}-\lambda p\_i=0$

$ (2)\frac{\partial L}{\partial x\_j} = \frac{a\_j}{ x\_j}-\lambda p\_j=0$

$ (3)\frac{\partial L}{\partial \lambda} = M-\sum\_{j=1}^np\_jx\_j=0$

from (1) and (2) it follows:

$ \frac{p\_i}{x\_i} = \frac{a\_i/x\_i}{a\_j/x\_j}=\frac{a\_ix\_j}{a\_jx\_i}$

$ x\_j = \frac{p\_ja\_jx\_i}{p\_ja\_j}$

$ x\_j$ into (3)

$ M = \sum\_{j=1}^np\_j(\frac{p\_ja\_jx\_i}{p\_j/a\_j})=0$

$ x\_i= \frac{a\_iM}{\sum\_{j=1}^na\_jp\_j}$

Furthermore I would like to interpret what happens if we have an efficency shock for good $i$. Meaning, good $i$ becomes cheaper. This leads to an increase in relative income. So $M$ increases which leads to an increased demand for good $x\_i$, rest hold constant. Is that correct?

**Tags:** utility, cobb-douglas, demand
> **Comments:**
> **username_0**: Sorry. Yes, a drop in $p_i$. And of course $M$ stays constant, but normally we say then that the relative income increases, since the household spends less money for each unit of $x_i$. So again, $p_i$ drops, this will increase the amount of $x_i$.  
> **username_1**: His derivation is incorrect (due to a few algebra mistakes). The Cobb-Douglas utility function leads to a set of demand functions in which demand for each good depends only on it's own price. Check the two-good case for reference. See my solution below.  
> **username_2**: @dlnB: You're right. I misread $p_j$ in the denominator as $p_i$.  

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## 1 answers

### Answer by username_1 on 2019-03-15. Score: 3 | Accepted answer

From (1) and (2) you get
$$\frac{x\_j}{x\_i}=\frac{a\_j p\_i}{a\_i p\_j},$$
or equivalently,
$$x\_j =\frac{a\_j p\_i}{a\_i p\_j} x\_i.$$
Substituting this into equation 3 for $j=2,...,n$ and $i=1$ (solving for the demand function for good 1) we get
$$M=p\_1x\_1 + \sum\_{j=2}^n p\_j \frac{a\_j p\_1}{a\_1 p\_j} x\_1$$
$$M=p\_1x\_1 + \sum\_{j=2}^n \frac{a\_j p\_1}{a\_1} x\_1$$
$$M=p\_1x\_1 + \frac{p\_1}{a\_1} x\_1\sum\_{j=2}^n a\_j $$
$$M=p\_1x\_1(1 + \frac{1}{a\_1}\sum\_{j=2}^n a\_j). $$
Finally, solving for $x\_1$ we get
$$x\_1^\* = \frac{M}{p\_1}(1+ \frac{\sum\_{j=2}^n a\_j}{a\_1})^{-1}$$
$$x\_1^\* =\frac{a\_1}{\sum\_{j=1}^n a\_j} \frac{M}{p\_1}.$$

Analogously,
$$x\_i^\*=\frac{a\_i}{\sum\_{j=1}^n a\_j} \frac{M}{p\_i},$$
for $i=1,...,n$.

As is always the case for demand functions derived from a Cobb-Douglas utility function, the consumer spends a constant share of income on each good. To see this, rearrange the previous equation to get
$$p\_ix\_i^\*=\frac{a\_i}{\sum\_{j=1}^n a\_j} M,$$
for $i=1,...,n$.


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