<p><span class="math-container">$I^j$</span> is just a known constant (i.e. we don't know anything about production, which would endogenously generate the amount of per capita income), but as you suggest the budget constraint for the consumer is that <span class="math-container">$\sum_i p_i d_i^j \leq I^j$</span> (here, you can assume this holds with equality).  </p>

<p>First, note that if <span class="math-container">$d_i^j &lt; \bar{d}_i$</span>, then the utility function will be negative. So, if we have sufficient income to afford the minimum consumption (i.e. <span class="math-container">$\sum_i p_i \bar{d}_i \leq I^j$</span>), to avoid this outcome the consumer will purchase at least <span class="math-container">$d_i^j \geq \bar{d}_i$</span> units of each good. Therefore, if we define <span class="math-container">$x_i^j \equiv d_i^j-\bar{d}_i$</span>, we know that <span class="math-container">$x_i^j \geq 0$</span>.</p>

<p>With this reformulation, we can rewrite this <em>Stone-Geary</em> utility function as a <em>Cobb Douglas</em> utility function, i.e.
<span class="math-container">$U^j = \prod_{i=1}^N (d_i^j - \bar{d}_i)^{\varphi_i}=\prod_{i=1}^N (x_i^j)^{\varphi_i}$</span>. Then, to solve for the optimal consumption of <span class="math-container">$x_i^j$</span>, you can just solve for optimal consumption with a Cobb Douglas utility function with a new income of <span class="math-container">$\bar{I}^j \equiv I^j - \sum_i p_i \bar{d}_i$</span> (that is, the remaining income left over after spending it on the minimum consumption level of each good). Once you have the optimal <span class="math-container">$x_i^j$</span>, the optimal consumption level of the good (inclusive of the minimum) is <span class="math-container">$d_i^j = x_i^j+\bar{d}_i$</span>.</p>

CC BY-SA 4.0
2018-12-13T21:10:39.307Z
2018-12-13T21:10:39.307Z
Jay
<p>$U^j = \prod^N_i \left(d^j_i-\bar{d}_i\right)^\oint$</p>

<p>where $d^j_i$ is the per capita consumption of good $i$ in any country $j$; $\bar{d}_i \geq 0$ denotes the minimum consumption of each good $i$, which are same across countries; $\displaystyle \sum_i \oint_i^j =1$. Assuming that per capita income $I^j$ is large enough to afford the minimum consumption. Derive the per capita demand for each good $i$ in country $j$.</p>

<p>I understand pi-product notation means it's a product function, but how do we differentiate a product function like this? I define my $I^j$ as $p_i d_i^j$; not sure if this is correct. Your help is much appreciated! Thank you in advance. :) </p>

CC BY-SA 3.0
2016-10-01T09:48:22.627Z
"Didn't work out" is somewhat vague. Computer ate your homework? Please at least include key words in the question like Heckscher-Ohlin trade model with non-homothetic preferences. Also it seems your question is more about differentiation than economics.
Giskard
CC BY-SA 3.0
2016-10-01T10:15:05.120Z
Hey @denesp, I was ashamed after reading your comment so I worked it out again; but I still couldn't add a superscript $i$ to $\oint$, please help if you don't mind. :)
Jojolim
CC BY-SA 3.0
2016-10-01T10:31:02.767Z
Not sure why you need that symbol but here you go. Aren't you looking for $\phi$ which is backslash phi?
Giskard
CC BY-SA 3.0
2016-10-01T10:32:15.577Z
@denesp I mean on the first equation (U equation), not the sum equation.  Thank you so much :)
Jojolim
CC BY-SA 3.0
2016-10-01T10:40:29.583Z
Did that as well.
Giskard
<international-trade>
Heckscher-Ohlin trade model with non-homothetic preferences
Jojolim
# Heckscher-Ohlin trade model with non-homothetic preferences

(Asked by: username_0 on 2016-10-01)

$U^j = \prod^N\_i \left(d^j\_i-\bar{d}\_i\right)^\oint$

where $d^j\_i$ is the per capita consumption of good $i$ in any country $j$; $\bar{d}\_i \geq 0$ denotes the minimum consumption of each good $i$, which are same across countries; $\displaystyle \sum\_i \oint\_i^j =1$. Assuming that per capita income $I^j$ is large enough to afford the minimum consumption. Derive the per capita demand for each good $i$ in country $j$.

I understand pi-product notation means it's a product function, but how do we differentiate a product function like this? I define my $I^j$ as $p\_i d\_i^j$; not sure if this is correct. Your help is much appreciated! Thank you in advance. :)

**Tags:** international-trade

> **Comments:**
> **username_1**: "Didn't work out" is somewhat vague. Computer ate your homework? Please at least include key words in the question like Heckscher-Ohlin trade model with non-homothetic preferences. Also it seems your question is more about differentiation than economics.  
> **username_0**: Hey @denesp, I was ashamed after reading your comment so I worked it out again; but I still couldn't add a superscript $i$ to $\oint$, please help if you don't mind. :)  
> **username_1**: Not sure why you need that symbol but here you go. Aren't you looking for $\phi$ which is backslash phi?  
> **username_0**: @denesp I mean on the first equation (U equation), not the sum equation.  Thank you so much :)  
> **username_1**: Did that as well.  

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## 1 answers

### Answer by username_2 on 2018-12-13. Score: 1

$I^j$ is just a known constant (i.e. we don't know anything about production, which would endogenously generate the amount of per capita income), but as you suggest the budget constraint for the consumer is that $\sum\_i p\_i d\_i^j \leq I^j$ (here, you can assume this holds with equality).

First, note that if $d\_i^j < \bar{d}\_i$, then the utility function will be negative. So, if we have sufficient income to afford the minimum consumption (i.e. $\sum\_i p\_i \bar{d}\_i \leq I^j$), to avoid this outcome the consumer will purchase at least $d\_i^j \geq \bar{d}\_i$ units of each good. Therefore, if we define $x\_i^j \equiv d\_i^j-\bar{d}\_i$, we know that $x\_i^j \geq 0$.

With this reformulation, we can rewrite this *Stone-Geary* utility function as a *Cobb Douglas* utility function, i.e.
$U^j = \prod\_{i=1}^N (d\_i^j - \bar{d}\_i)^{\varphi\_i}=\prod\_{i=1}^N (x\_i^j)^{\varphi\_i}$. Then, to solve for the optimal consumption of $x\_i^j$, you can just solve for optimal consumption with a Cobb Douglas utility function with a new income of $\bar{I}^j \equiv I^j - \sum\_i p\_i \bar{d}\_i$ (that is, the remaining income left over after spending it on the minimum consumption level of each good). Once you have the optimal $x\_i^j$, the optimal consumption level of the good (inclusive of the minimum) is $d\_i^j = x\_i^j+\bar{d}\_i$.


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