<p>yes. Notice that a good is normal, inferior Giffen for a certain value of prices/income. For example, a good might be inferior at some prices/income but normal for some other values of the prices/income.</p>
<p>In some cases, however, you can make stronger statments. For example, homothetic preferences always lead to normal demands as
<span class="math-container">$$
x_1(p_1, p_2,m) = x_1(p_1, p_2,1) m 
$$</span>
where <span class="math-container">$x_1(p_1, p_2,1)$</span> is the unit income demand function which is always greater or equal to zero.</p>
<p>From this, it follows that homothetic preference can never lead to Giffen goods (but to show this, you do need the Slutsky decomposition).</p>

CC BY-SA 4.0
2021-05-11T15:58:49.813Z
Awesome, thanks. Yeah it's a bit unfortunate I can't use the Slutsky decomposition. I might come back with another question about this once I learnt about that haha.
j3141592653589793238
CC BY-SA 4.0
2021-05-11T15:53:58.690Z
2021-05-11T15:53:58.690Z
tdm
<p>Say I've got a function <span class="math-container">$x_1(p_1,p_2,m)$</span> where <span class="math-container">$p_1, p_2$</span> are the prices for good 1, good 2 respectively and m is the income.</p>
<p>Now, I haven't heard of the Slutsky equation yet nor the income/substitution effect.</p>
<p>How do I show for this function that it is an inferior/normal and ordinary/Giffen good?</p>
<p>My initial thought was to simply use the partial derivate and to check whether it's greater or less than zero, since</p>
<p><span class="math-container">$\frac{\partial x_1(...)}{\partial m} \gt 0$</span> should be true for normal goods, <span class="math-container">$\frac{\partial x_1(...)}{\partial m} \lt 0$</span> for inferior goods, <span class="math-container">$\frac{\partial x_1(...)}{\partial p_1} \gt 0$</span> for Giffen goods and <span class="math-container">$\frac{\partial x_1(...)}{\partial p_1} \lt 0$</span> for ordinary goods.</p>
<p>Is this correct?</p>
<p>Is this all I can do to formally show that the good is normal/ordinary? Is this sufficient?</p>
<p>Also, if I can show that the same function for <span class="math-container">$x_1(m)$</span> is homogeneous of degree 1 (i.e. homothetic), i.e. <span class="math-container">$tx_1(m)=x_1(tm)$</span>, this means that the partial derivative <span class="math-container">$\frac{\partial x_1(...)}{\partial m}$</span> is homogeneous of degree 0, so it would be a constant. Could I thus conclude that the good is strictly normal/inferior (is that a term?), i.e. it doesn't change if the income changes?</p>

CC BY-SA 4.0
2021-05-11T15:31:28.197Z
yes. Notice that a good is normal, inferior Giffen for a certain value of prices/income. For example, a good might be inferior at some prices/income but normal for some other values of the prices/income.
tdm
CC BY-SA 4.0
2021-05-11T15:34:36.167Z
You're right; however if I can show that the partial derivative $\partial x_1(...)/\partial m$ is not dependent on $m$, that means it's a constant and the function $x_1$ is homothetic. So for homothetic functions this shouldn't matter? @tdm
j3141592653589793238
CC BY-SA 4.0
2021-05-11T15:49:35.940Z
@tdm Please post answers as answers.
Giskard
<microeconomics><mathematical-economics><utility><elasticity>
Without knowing the Slutsky equation and income/substitution effect, how can I show a certain good is inferior or Giffen?
j3141592653589793238
# Without knowing the Slutsky equation and income/substitution effect, how can I show a certain good is inferior or Giffen?

(Asked by: username_0 on 2021-05-11)

Say I've got a function $x\_1(p\_1,p\_2,m)$ where $p\_1, p\_2$ are the prices for good 1, good 2 respectively and m is the income.

Now, I haven't heard of the Slutsky equation yet nor the income/substitution effect.

How do I show for this function that it is an inferior/normal and ordinary/Giffen good?

My initial thought was to simply use the partial derivate and to check whether it's greater or less than zero, since

$\frac{\partial x\_1(...)}{\partial m} \gt 0$ should be true for normal goods, $\frac{\partial x\_1(...)}{\partial m} \lt 0$ for inferior goods, $\frac{\partial x\_1(...)}{\partial p\_1} \gt 0$ for Giffen goods and $\frac{\partial x\_1(...)}{\partial p\_1} \lt 0$ for ordinary goods.

Is this correct?

Is this all I can do to formally show that the good is normal/ordinary? Is this sufficient?

Also, if I can show that the same function for $x\_1(m)$ is homogeneous of degree 1 (i.e. homothetic), i.e. $tx\_1(m)=x\_1(tm)$, this means that the partial derivative $\frac{\partial x\_1(...)}{\partial m}$ is homogeneous of degree 0, so it would be a constant. Could I thus conclude that the good is strictly normal/inferior (is that a term?), i.e. it doesn't change if the income changes?

**Tags:** microeconomics, mathematical-economics, utility, elasticity
> **Comments:**
> **username_1**: yes. Notice that a good is normal, inferior Giffen for a certain value of prices/income. For example, a good might be inferior at some prices/income but normal for some other values of the prices/income.  
> **username_0**: You're right; however if I can show that the partial derivative $\partial x_1(...)/\partial m$ is not dependent on $m$, that means it's a constant and the function $x_1$ is homothetic. So for homothetic functions this shouldn't matter? @tdm  
> **username_2**: @tdm Please post answers as answers.  

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## 1 answers

### Answer by username_1 on 2021-05-11. Score: 3 | Accepted answer

yes. Notice that a good is normal, inferior Giffen for a certain value of prices/income. For example, a good might be inferior at some prices/income but normal for some other values of the prices/income.

In some cases, however, you can make stronger statments. For example, homothetic preferences always lead to normal demands as
$$
x\_1(p\_1, p\_2,m) = x\_1(p\_1, p\_2,1) m
$$
where $x\_1(p\_1, p\_2,1)$ is the unit income demand function which is always greater or equal to zero.

From this, it follows that homothetic preference can never lead to Giffen goods (but to show this, you do need the Slutsky decomposition).

> **Comments:**
> **username_0**: Awesome, thanks. Yeah it's a bit unfortunate I can't use the Slutsky decomposition. I might come back with another question about this once I learnt about that haha.  

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